MANION, Circuit Judge.

Defendant David Carmel pleaded guilty to one count of possessing an unregistered machine gun in violation of 26 U.S.C. § 5861, reserving his right to appeal from the district court's denial of his motion to suppress, its denial of his motion for a hearing pursuant to Franks v. Delaware, 438 U.S. 154, 98 S.Ct. 2674, 57 L.Ed.2d 667 (1978), and its denial of his challenge to the constitutionality of 26 U.S.C. § 5861. We AFFIRM.

I.

David Carmel was being investigated by Immigration and Customs Enforcement ("ICE") and Defense Criminal Investigative Service ("DCIS"), an arm of the Department of Defense. ICE and DCIS believed that Carmel had stolen and sold military supplies and weapons. During a phone conversation between Carmel and an undercover ICE agent, Carmel indicated that he had three firearms — a "Rheinmetall MG3, a MG 34, and a Heckler Kock HK21." Carmel invited the agent to shoot with him at his home and added that whatever ammunition the agent would bring, he owned a weapon that would fire it.

On May 30, 2007, DCIS agents arrested Carmel in his car. A local sheriffs investigator, Chad Holum, was present during the arrest and found in Carmel's car two top handles that could potentially be used in an M-16 machine gun. Holum then applied in Wisconsin state court for a warrant to search Carmel's home. The application for the warrant contained an affidavit and an accompanying statement explaining why Holum believed probable cause existed to search Carmel's home. The statement first recounted the conversation between Carmel and the undercover agent. Next, the statement noted the discovery of the two top handles, which were described as "made for a M16 machine gun." Holum's statement further indicated that based on his experience in the Marine Corps and as a law enforcement officer, top handles were necessary to fire an M-16. The statement noted that in a post-arrest interview, Carmel had said that he possessed one machine gun, an "MG M-119," that was properly registered with the federal government. The statement continued: "When asked about other types of machine guns, Carmel refused to answer any questions in this area." Carmel also admitted possessing other machine gun barrels and parts. Finally, the statement noted that Carmel had told his father in a recorded telephone conversation that he had been arrested by federal authorities, and that "his father could guess what it is for." The affidavit in support of the search warrant identified the three weapons — the Rheinmetall MG3, the MG 34, and the Heckler Koch HK2 — as machine guns.

The state court issued the search warrant, permitting officers to search for the three identified weapons. When officers searched Carmel's home and property, they discovered several rifles and a rocket launcher, and the officers notified federal authorities. Field-testing of numerous other weapons revealed approximately thirty machine guns. The weapons were not seized immediately. Instead, officers sought to secure a warrant to seize the weapons as well as other parts and computers that could contain evidence of illegal weapon manufacturing and sales.

A federal search warrant was subsequently issued to seize the machine guns as well as other evidence of criminal activity. After more than sixty machine guns were recovered, Carmel was charged with violations of 18 U.S.C. § 922(o), which provides that "[e]xcept as provided in paragraph (2), it shall be unlawful for any person to transfer or possess a machinegun." However, a subsequent grand jury indictment charged Carmel with violating 26 U.S.C. § 5861(d), which provides that "[i]t shall be unlawful for any person . . . to receive or possess a firearm which is not registered to him in the National Firearms Registration and Transfer Record."

Carmel moved to suppress the machine gun evidence seized at his home and requested a Franks hearing to determine whether Holum recklessly omitted material facts from his affidavit. Carmel also moved to dismiss the indictment, arguing that 26 U.S.C. § 5861(d) had been implicitly repealed by the enactment of 18 U.S.C. § 922(d), that Congress lacked a constitutional basis to enact 26 U.S.C. § 5861(d), and that § 5861(d) violated the Due Process Clause because it was impossible to comply with and was unconstitutionally vague.

The magistrate judge denied Carmel's motion for a Franks hearing. Then, the magistrate judge issued a report in which he recommended that the district court deny the motions to dismiss the indictment and suppress the evidence. Subsequently, Carmel and the government entered into a plea agreement under which Carmel would plead guilty to one count of violating 26 U.S.C. § 5861(d); however, Carmel reserved his right to withdraw the plea should the district court rule in his favor on the motions and to appeal should the district court rule against him. The district court then issued an order adopting the recommendation of the magistrate judge and denying the motions to dismiss the indictment and to suppress the machine guns. Carmel was sentenced to forty-six months in prison and thirty-six months of supervised release. Carmel appeals.

II. A.

On appeal, Carmel first argues that the state search warrant was not supported by probable cause and that the resulting federal search warrant constitutes noxious "fruit of the poisonous tree." United States v. Grogg, 534 F.3d 807, 810 (7th Cir. 2008). "A search warrant affidavit establishes probable cause when, based on the totality of the circumstances, it sets forth sufficient evidence to induce a reasonably prudent person to believe that a search will uncover evidence of a crime." United States v. Curry, 538 F.3d 718, 729 (7th Cir. 2008). The judge issuing the warrant "need only conclude that it would be reasonable to seek the evidence in the place indicated in the affidavit," id., but the judge "may not rely solely upon conclusory allegations or a bare bones affidavit when issuing a warrant." Id. When reviewing a search that was conducted pursuant to a warrant, "we must afford great deference to the issuing judge's conclusion" that probable cause to search existed. United States v. Prideaux-Wentz, 543 F.3d 954, 958 (7th Cir. 2008). "Judges may draw reasonable inferences from the totality of the circumstances in determining whether probable cause exists to issue a warrant." Id. at 961.

In this case, probable cause existed to search Carmel's home. Carmel claimed to have three machine guns at his home in a conversation with an undercover agent, but later told police that he possessed one properly licensed machine gun. Moreover, the single properly licensed machine gun was not identical to the three machine guns Carmel earlier claimed to possess. It is reasonable to infer from this evidence that Carmel possessed three unlicensed, illegal machine guns. Coupled with the top handles that could be used in an M-16 machine gun, as well as Carmel's admission that he possessed other machine gun parts, a reasonably prudent person could conclude that, if police searched Carmel's home, they would discover several illegal machine guns. Accordingly, the state search warrant was supported by probable cause.

Carmel attacks the affidavit piece by piece. First, he argues that the affidavit simply makes a "bald assertion" that the Rheinmetall MG3, MG 34, and Heckler Kock HK21 were machine guns. Although Carmel acknowledges that the affidavit mentions Holum's experience in the Marine Corps and as a law enforcement officer, Carmel argues that this experience is "unexplained." Certainly, Holum could have more clearly set forth his experience with machine guns; however, perfection is not required for an affidavit to pass constitutional muster. Rather, the affidavit only must be sufficient to allow a reasonably prudent person to believe that evidence of a crime will be found. Curry, 538 F.3d at 729. A reasonably prudent person could believe that a police officer and former Marine would be able to identify a machine gun.

Carmel then argues that because there were various exceptions to the state law at issue (for instance, Carmel could have lawfully possessed a machine gun "not usable as a weapon and possessed as a curiosity, ornament or keepsake," Wis. STAT. § 941.27(2)), the affidavit should have indicated that Carmel's weapons did not fit into any of the specified categories. However, the existence of limited statutory exceptions to the general prohibition on machine guns does not prevent a reasonable person from inferring that Carmel's possession of the machine guns was illegal. See United States v. Decoteau, 932 F.2d 1205, 1207 n. 1 (7th Cir. 1991) (noting that, despite the existence of a narrow statutory exception permitting legal possession, witnessing a person in possession of a sawed-off shotgun gives rise to probable cause to believe that the firearm is unregistered).

Similarly, Carmel notes that the top handles found in his car, although usable in an M-16, could also be used in an AR-15, a legal weapon. However, despite the term "probable," finding probable cause "demands even less than probability." Woods v. City of Chicago, 234 F.3d 979, 996 (7th Cir. 2000). To show probable cause "requires more than bare suspicion but need not be based on evidence sufficient to support a conviction, nor even a showing that the officer's belief is more likely true than false." Id. The potential lawful use of the top handles does not negate the other incriminatory inference, and in light of the other evidence pointing toward illegal possession, the top handles provide further support for a finding of probable cause.

Finally, Carmel argues that his statement that he possessed one machine gun legally cannot be used to infer that he possessed other machine guns illegally, because to do so would use his silence against him in violation of Miranda v. Arizona, 384 U.S. 436, 86 S.Ct. 1602, 16 L.Ed.2d 694 (1966). The magistrate judge agreed with Carmel, concluding that "[i]t would have been improper for the state court to infer from Carmel's invocation of his right to remain silent a tacit admission that his other machine guns were unlicensed." The government concedes that use of Carmel's silence in the probable cause calculus would be improper. However, it is not Carmel's silence that created an inference of criminal activity, but his statements to the undercover agent. From the statement that he possessed three machine guns and his later statement that he legally possessed one machine gun, a reasonable inference may be drawn that some machine guns were unlawfully possessed. Neither the Fifth Amendment nor Miranda prevents police officers from using two contradictory statements by a defendant when determining probable cause. See Anderson v. Charles, 447 U.S. 404, 408, 100 S.Ct. 2180, 65 L.Ed.2d 222 (1980) (holding that inconsistent post-Miranda statements could be used against a defendant despite prohibition on use of silence, because regarding "the subject matter of his statements, the defendant has not remained silent at all"); United States v. Santiago, 428 F.3d 699, 704 (7th Cir. 2005) (noting that a prosecutor may comment upon inconsistencies in a defendant's post-Miranda statements without violating Miranda).

For these reasons, the affidavit and accompanying statement gave rise to probable cause. Moreover, even if probable cause did not exist, the evidence seized should not be excluded. "[Suppression of evidence seized pursuant to a search warrant that is later declared invalid is inappropriate if the officers who executed the warrant relied in good faith on the issuing judge's finding of probable cause." United States v. Watts, 535 F.3d 650, 656-57 (7th Cir. 2008) (citing United States v. Leon, 468 U.S. 897, 924, 104 S.Ct. 3405, 82 L.Ed.2d 677 (1984)). "A defendant can rebut the presumption of good faith by showing, as relevant here, that the supporting affidavit is so facially deficient that no reasonable officer could have relied upon it." Id. at 657 (citing Leon, 468 U.S. at 922-23, 104 S.Ct. 3405). In light of the discrepancy between Carmel's statements that he had one legally registered machine gun but possessed three, Holum's reference to his military and law enforcement background to show knowledge of machine guns, and the top handles found in Carmel's car, a reasonable officer could have relied on the magistrate's finding that probable cause existed. Accordingly, we affirm the district court's refusal to suppress the evidence.

B.

Next, Carmel argues that the district court erred in denying his motion for a Franks hearing. In Franks v. Dela-ware, the United States Supreme Court held that under limited circumstances a defendant may be entitled to a hearing to challenge the truth of statements made in a search warrant affidavit. United States v. Souffront, 338 F.3d 809, 822 (7th Cir. 2003). The defendant must make a "substantial preliminary showing" that a false statement was made either intentionally or recklessly. Id. (citing Franks, 438 U.S. at 155-56, 98 S.Ct. 2674). A Franks hearing is also required if a defendant shows that "the affiant intentionally or recklessly omitted material information." United States v. Hoffman, 519 F.3d 672, 675 (7th Cir. 2008) (emphasis added). However, if probable cause to issue the warrant would still exist even if the false statement or material omission were corrected, then no Franks hearing is required. Souffront, 338 F.3d at 822 (citing Franks, 438 U.S. at 171-72, 98 S.Ct. 2674). A district court's decision to grant or deny a Franks hearing is reviewed for clear error. Hoffman, 519 F.3d at 675.

In this case, Carmel sought a Franks hearing, contending that Holum's statement omitted two material facts. First, he claims the statement should have indicated that the top handles could be used legally in an AR-15, as well as in an M-16. Second, Carmel complains that Holum failed to note that he had registered a machine gun with the Bureau of Alcohol, Tobacco, and Firearms ("ATF"), in addition to possessing a license for the weapon. However, Carmel has not shown that a Franks hearing is required. With regard to the top handles, even if Holum had noted that the top handles could be used in an AR-15, that would not negate the fact that the top handles could also be used in an M-16. In light of the other evidence indicating that Carmel possessed illegal machine guns, the potential alternate use of the top handles would not have prevented the conclusion that probable cause to search existed.

Camel's argument regarding registration with the ATF is also meritless. The statement submitted in support of the search warrant implied that Carmel possessed one machine gun legally. Adding that the gun was registered with the ATF would have merely underscored the point already made. Accordingly, even if that fact had been included, the probable cause analysis would remain the same. For these reasons, we find no clear error in the district court's denial of the motion for a Franks hearing.

C.

Finally, Carmel attacks the constitutional and statutory basis for the indictment under 26 U.S.C. § 5861(d). That section provides that "[i]t shall be unlawful for any person . . . to receive or possess a firearm which is not registered to him in the National Firearms Registration and Transfer Record." 26 U.S.C. § 5861(d). Carmel argues that this statute, which was enacted in 1968, was implicitly repealed by the enactment of 18 U.S.C. § 922(o) in 1986. Section 922(o) states:

(1) Except as provided in paragraph (2), it shall be unlawful for any person to transfer or possess a machinegun.

(2) This subsection does not apply with respect to —

(A) a transfer to or by, or possession by or under the authority of, the United States or any department or agency thereof or a State, or a department, agency, or political subdivision thereof; or

(B) any lawful transfer or lawful possession of a machinegun that was lawfully possessed before the date this subsection takes effect.

Carmel argues that because he could not lawfully possess the machine guns under § 922(o), compliance with § 5861(d) was impossible. In other words, Carmel insists that he could not register the machine gun if he could not legally possess it in the first place. Thus, according to Carmel, because ownership of a machine gun was made impossible by § 922(o), Congress must have intended to repeal the registration requirement of § 5861(d). Furthermore, Carmel contends that his conviction for failing to take an action that by law he is forbidden from doing violates due process.

The circuits have split on this issue. The Tenth Circuit has agreed with Carmel, finding that § 922(o) implicitly repealed § 5861(d). See United States v. Dalton, 960 F.2d 121 (10th Cir. 1992). Dalton concluded that a conviction under § 5861(d) violated fundamental fairness, stating: "Because the crimes of which Dalton was convicted thus have as an essential element his failure to do an act that he is incapable of performing, his fundamental fairness argument is persuasive." Id. at 124. Dalton further held that § 5861(d) no longer had a constitutional basis:

[B]ecause the registration requirements of the National Firearms Act were passed pursuant to the taxing power . . . and because after the enactment of section 922(o) the government will no longer register or tax machineguns, . . . section 922(o) has "removed the constitutional legitimacy of registration as an aid to taxation[.]"

Id. at 124-25 (citations omitted) (quoting United States v. Rock Island Armory, 773 F.Supp. 117, 125 (C.D.I11.1991)).

The Fourth Circuit rejected Dalton in United States v. Jones, 976 F.2d 176 (4th Cir. 1992). Jones stated, "[s]imply put, Jones can comply with both acts by refusing to deal in newly-made machine guns." Id. at 183. The Jones court further elaborated:

What Jones is really complaining about is that the amendment to the Gun Control Act effectively rendered possession of certain guns automatic violations of both the Gun Control Act and the National Firearms Act. Yet there is nothing either inconsistent or unconstitutionally unfair about Congress' decision to do so. And, faced with two equally applicable penal statutes, there is nothing wrong with the government's decision to prosecute under one and not the other, so long as it does not discriminate against any class of defendants, which Jones does not allege.

Id. Jones also concluded that a constitutional basis continues to exist for § 5861(d), because "[n]otwithstanding the effective ban on machine guns made after 1986, the making of even illegal machine guns continues to be taxed." Id. Moreover, "knowing the chain of possession and transfer assists in determining who made the firearm and hence is `supportable as in aid of a revenue purpose.'" Id. at 184 (quoting Sonzinsky v. United States, 300 U.S. 506, 513, 57 S.Ct. 554, 81 L.Ed. 772 (1937)).

The Third, Fifth, Sixth, Eighth, Ninth, and Eleventh Circuits have all followed Jones. United States v. Grier, 354 F.3d 210, 214 (3d Cir. 2003); United States v. Bournes, 339 F.3d 396, 399 (6th Cir. 2003); United States v. Elliott, 128 F.3d 671, 672 (8th Cir. 1997); Hunter v. United States, 73 F.3d 260, 261-62 (9th Cir. 1996); United States v. Rivera, 58 F.3d 600, 601-02 (11th Cir. 1995); United States v. Ardoin, 19 F.3d 177, 179-80 (5th Cir. 1994). Indeed, we previously adopted the reasoning in Jones in United States v. Ross, 9 F.3d 1182, 1193-94 (7th Cir. 1993). However, Ross was subsequently vacated on other grounds, 511 U.S. 1124, 114 S.Ct. 2129, 128 L.Ed.2d 860 (1994), and hence is no longer precedentially binding.

We again adopt the reasoning of the majority of the circuits as set forth in Jones. Section 922(o) is reconcilable with § 5861(d): Carmel could have complied with both statutes simply by declining to possess sixty illegal machine guns. Congress may lawfully punish the same action under two separate statutes without running afoul of the Due Process Clause. See United States v. Malik, 385 F.3d 758, 760 (7th Cir. 2004) (stating that "[w]hen the same acts violate multiple laws, the prosecutor is free to choose the one with the highest sentence") (citing United States v. Batchelder, 442 U.S. 114, 123-24, 99 S.Ct. 2198, 60 L.Ed.2d 755 (1979)). Moreover, Congress could lawfully require registration under the taxing power, because illegally made machine guns are still subject to taxation. Jones, 976 F.2d at 183 (citing 26 U.S.C. § 5821). Accordingly, the district court properly denied Carmel's motion to dismiss the indictment.

III

Because the state search warrant was supported by probable cause, the district court properly denied Carmel's motion to suppress. Moreover, because Carmel has not shown the existence of an intentional or reckless omission from the accompanying affidavit and statement, the district court did not err in denying the motion for a Franks hearing. Finally, we agree with the majority of our sister circuits that 18 U.S.C. § 922( o) is reconcilable with 26 U.S.C. § 5861(d), and therefore conclude that the district court properly denied Carmel's motion to dismiss the indictment.

We AFFIRM.