Appellant Broncheau, an Indian, was indicted for assault resulting in serious bodily injury of a non-Indian within Indian country, in violation of 18 U.S.C. §§ 1153 and 113(f).

Broncheau moved to dismiss the indictment, alleging that: (1) the district court lacked jurisdiction because the tribal courts have exclusive jurisdiction to punish Indians for crimes committed on Indian land; and (2) § 1153 is unconstitutional because the disparity in treatment of Indians and non-Indians for the same offense denied Broncheau’s right to due process and equal protection. At the hearing on the motion to dismiss, Broncheau additionally alleged that the indictment was jurisdictionally defective because it did not allege that Broncheau was an enrolled Indian. He further argued that if enrollment did not have to be pleaded, § 1153 was void for vagueness.

Upon the district court’s denial of the motion to dismiss and pursuant to a plea bargaining agreement, Broncheau entered a plea of guilty to the lesser included misdemeanor offense of simple assault, in violation of 18 U.S.C. §§ 1153 and 113(e). In entering his plea of guilty, Broncheau admitted that he was an enrolled Nez Perce Indian and had struck and assaulted Michael Cram, a non-Indian, within the boundaries of the Nez Perce Indian Reservation. The district court accepted. Broncheau’s guilty plea and subsequently fined him $150.00. Broncheau now challenges as erroneous the district court’s denial of his motion to dismiss. We reject Broncheau’s arguments as not meritorious and affirm.

I.

Sufficiency of the Indictment

Broncheau contends that federal jurisdiction pursuant to 18 U.S.C. § 1153 was lacking because the indictment did not allege that he was an “enrolled” Indian.

The indictment incorporated the language of § 1153 in setting out the necessary elements to be proved by the Government: Broncheau was an Indian who had stabbed a non-Indian within the boundaries of the Nez Perce Indian Reservation. Unless § 1153 is void for vagueness, see part II infra, the indictment is sufficient because it closely follows the language of the statute. See United States v. Anderson, 532 F.2d 1218, 1222 (9th Cir. 1976), cert. denied, 429 U.S. 839 , 97 S.Ct. 111 , 50 L.Ed.2d 107 (1976).

Moreover, although an allegation of enrollment may be sufficient for purposes of alleging federal jurisdiction, see e. g., United States v. Heath, 509 F.2d 16, 20 (9th Cir. 1974), enrollment has not yet been held to be an absolute requirement of federal jurisdiction. See United States v. Antelope, 430 U.S. 641 , 646 n. 7, 97 S.Ct. 1395 , 51 L.Ed.2d 701 (1977). Nor should it be. Enrollment is the common evidentiary means of establishing Indian status, but it is not the only means nor is it necessarily determinative. See United States v. Indian Boy X, 565 F.2d 585, 594 (9th Cir. 1977), cert. denied, - U.S. -, 99 S.Ct. 131 , 58 L.Ed.2d 139 (1978); United States v. Ives, 504 F.2d 935, 953 (9th Cir. 1974), vacated on other grounds, 421 U.S. 944 , 95 S.Ct. 1671 , 44 L.Ed.2d 97 (1975); Ex parte Pero, 99 F.2d 28, 30 (7th Cir. 1938); F. Cohen, Handbook of Federal Indian Law 2-5 (1942).

An indictment must state the essential facts to apprise the defendant of the charges against him so that the defendant may prepare a defense, but the indictment may be construed to include facts necessarily implied by the allegations. United States v. Anderson, 532 F.2d at 1222 ; United States v. Heath, 509 F.2d at 20 . The purpose of an indictment was served here. Broneheau had adequate notice of his classification as an Indian. The allegation encompassed by implication those facts material to proving that an individual is an Indian, e. g., enrollment, blood quantum, residence.

Alleging that Broneheau was an Indian was sufficient for purposes of federal jurisdiction under § 1153. Further refinement was not required.

II.

Vagueness of Section 1153

Broneheau argues that § 1153 is impermissibly vague because it does not define the term “Indian” and thereby permits arbitrary prosecutorial discretion in the prosecution of individuals for § 1153 crimes.

“It is well established that vagueness challenges to statutes which do not involve First Amendment freedoms must be examined in the light of the facts of the case at hand.” United States v. Mazurie, 419 U.S. 544, 550 , 95 S.Ct. 710, 714 , 42 L.Ed.2d 706 (1975) (citation omitted); accord, United States v. Louderman, 576 F.2d 1383, 1388 (9th Cir. 1978), cert. denied,U.S. -, 99 S.Ct. 257 , 58 L.Ed.2d 243 (1978); United States v. Ocegueda, 564 F.2d 1363, 1365 (9th Cir. 1975). Thus, the test is not whether the statute is vague in the abstract but whether it is vague as applied in the particular circumstances of the case. Id. Moreover, if judicial explication makes a statute clear so that fair notice is afforded, vagueness may not be imputed. United States v. Fithian, 452 F.2d 505 , 506 n. 1 (9th Cir. 1971).

Unlike the term “Indian country,” which has been defined in 18 U.S.C. § 1151 , the term “Indian” has not been statutorily defined but instead has been judicially explicated over the years. The test, first suggested in United States v. Rogers, 45 U.S. 567 , 11 L.Ed. 1105 (1845), and generally followed by the courts, considers (1) the degree of Indian blood; and (2) tribal or governmental recognition as an Indian. United States v. Dodge, 538 F.2d 770, 786 (8th Cir. 1976), cert. denied, 429 U.S. 1099 , 97 S.Ct. 1118 , 51 L.Ed.2d 547 (1977) (enrollment and one-fourth Indian blood); F. Cohen, Handbook of Federal Indian Law 3 (1942); see United States v. Indian Boy X, 565 F.2d at 594 (enrollment and residence); United States v. Lossiah, 537 F.2d 1250, 1251 (4th Cir. 1976) (enrollment and three-fourths Indian blood); Azure v. United States, 248 F.2d 335, 337 (8th Cir. 1957) (enrollment).

We therefore believe that the term “Indian,” as judicially developed from 1845 to the present, “has a meaning sufficiently precise for a man of average intelligence to ‘reasonably understand that his contemplated conduct is proscribed.’ ” United States v. Mazurie, 419 U.S. at 553 , 95 S.Ct. at 715-716 . Moreover, we note that Broncheau admitted that he was an enrolled Indian at the time his guilty plea was entered and has never suggested that he did not understand the term “Indian” as it applied to him. In addition, the record shows that the district judge, who had lived in the community and was acquainted with the Broncheau family, identified Broncheau as an Indian. As in United States v. Mazurie, the distinction between Indians and non-Indians was commonly understood and readily made by all concerned. Id. at 553 n. 10, 95 S.Ct. 710 .

Contrary to Broncheau’s assertion, merely because the term “Indian” has been judicially defined on a case-to-case basis does not render § 1153 impermissibly vague as applied to him. The standard for determining who is an Indian has been adequately established by judicial decision. The prosecution of Broncheau was neither arbitrary nor irrational under the circumstances.

III.

Federal Jurisdiction of Section 1153 Crimes

Broncheau asserts that tribal courts have exclusive jurisdiction over crimes committed by Indians within Indian country. To hold otherwise, Broncheau argues, would be contrary to the purpose of the Indian Crimes Act of 1976. ( Pub.L. 94-297, 90 Stat. 585, 94th Cong. S. 2129, amending § 1153.)

The sovereignty of Indian tribes “exists only at the sufferance of Congress and is subject to complete defeasance.” United States v. Wheeler, 435 U.S. 313, 323 , 98 S.Ct. 1079, 1086 , 55 L.Ed.2d 303 (1978). As sovereign entities, however, Indian tribes generally retain their inherent sovereign powers until divested of them by treaty or statute. Id.; Ortiz-Barraza v. United States, 512 F.2d 1176, 1179 (9th Cir. 1975). Included in those inherent powers is the power to enact and enforce penal laws against tribal members. Thus, in Ex parte Crow Dog, 109 U.S. 556 , 3 S.Ct. 396 , 27 L.Ed. 1030 (1883), the Supreme Court concluded that an Indian tribe retained exclusive jurisdiction to punish an Indian for the murder of another Indian in the absence of any congressional direction to the contrary. In response to that decision, Congress enacted the Major Crimes Act, 18 U.S.C. § 1153 , to confer jurisdiction on the federal courts to punish certain offenses committed by Indians in Indian country because “tribal remedies were either nonexistent or incompatible with principles that Congress thought should be controlling.” Keeble v. United States, 412 U.S. 205, 210 , 93 S.Ct. 1993, 1996 , 36 L.Ed.2d 844 (1973).

As Felix Cohen noted, “[ajlthough [the Major Crimes Act] does not expressly terminate tribal jurisdiction over the enumerated crimes, and may be interpreted as conferring only a concurrent jurisdiction upon the federal courts, it is arguable that the statute removes all jurisdiction over the enumerated crimes from the Indian authorities.” F. Cohen, Handbook of Federal Indian Law 147 (1942). Unlike the General Crimes Act, 18 U.S.C. § 1152 , where Congress continued to recognize the power of Indian tribes to punish offenses committed by tribal members, § 1153 does not contain any exceptions to federal jurisdiction for the offenses enumerated therein. United States v. Wheeler, 435 U.S. at 325 n. 22, 98 S.Ct. 1079 .

Contrary to Broncheau’s assertion, the Indian Crimes Act of 1976 offers no support for his proposition that tribal jurisdiction is exclusive for major crimes committed by Indians in Indian country. As the legislative history demonstrates, the Act in fact further solidified federal jurisdiction over certain offenses. First, to cor rect a constitutional defect precluding federal jurisdiction, the Act required that the crimes of rape, assault with intent to commit rape, assault with a dangerous weapon, and assault resulting in serious bodily injury be defined and punished in accordance with federal law. Second, it expanded the list of enumerated offenses to include kidnapping because of the inadequacy of the maximum penalty (6 months imprisonment and a $500 fine, see 25 U.S.C. § 1302 (1)) that tribal courts could impose on tribal members for any offense. Finally, it accepted judicial interpretation of § 1152 that federal jurisdiction existed over other “peculiarly federal” offenses committed by Indians in Indian country despite the exception to jurisdiction provided for in § 1152. See H.R.Rep.No.1038, 94th Cong., 2d Sess., reprinted in [1976] U.S.Code Cong. & Admin.News, p. 1125. Moreover, Congress specifically noted that this legislation did not address the equal protection issue raised here. Id. at 1127 n. 2.

The unsettled question then is not whether tribal jurisdiction is exclusive, but whether federal jurisdiction under the Major Crimes Act is exclusive or concurrent with tribal jurisdiction. See Oliphant v. Suquamish Indian Tribe, 435 U.S. 191 , 203 n. 14, 98 S.Ct. 1011 , 55 L.Ed.2d 209 (1978); United States v. Wheeler, 435 U.S. at 325 n. 22, 98 S.Ct. 1079 . We find that Congress in the exercise of its plenary power has deprived tribal courts of exclusive jurisdiction over § 1153 offenses. Tribal courts may have concurrent jurisdiction but we have no reason to decide that question today. See United States v. Wheeler, 435 U.S. at 329-332, 98 S.Ct. 1079 . It is sufficient for this appeal that the district court had jurisdiction to try Broncheau.

IV.

Denial of Equal Protection and Due Process

Broncheau maintains that the assertion of federal jurisdiction for § 1153 crimes is based on an impermissible racial classification and denies him equal protection and due process of the law because, as an Indian, he is subject to harsher standards than non-Indians in the same situation.

Broncheau fails to note that, except for the offense committed — assault instead of murder — his position is identical to that of the defendants in United States v. Antelope, 430 U.S. 641 , 97 S.Ct. 1395 , 51 L.Ed.2d 701 (1977). In Antelope , the Supreme Court rejected the argument that § 1153 was based upon an impermissible racial classification because federal regulation was

430 U.S. at 646 , 97 S.Ct. at 1399 (citation omitted). The Court also concluded that the disparity between federal and state law was irrelevant as long as federal law was evenhanded. The Court found that the Indians in Antelope were subject to the same body of law as any other individual charged with the same offense committed in a federal enclave. 430 U.S. at 648 , 97 S.Ct. 1395 . So was Broncheau. Thus, federal law does not create any disparity by providing for different penalties and burden of proof, as was the situation found unconstitutional in United States v. Cleveland, 503 F.2d 1067 (9th Cir. 1974) and United States v. Big Crow, 523 F.2d 955 (8th Cir. 1975), cert. denied, 424 U.S. 920 , 96 S.Ct. 1126 , 47 L.Ed.2d 327 (1976).

Broncheau was treated by the Federal Government “in the same manner as all persons within federal jurisdiction pursuant to a regulatory scheme that did not erect impermissible racial classifications.” United States v. Antelope, 430 U.S. at 649-50 , 97 S.Ct. at 1400 . Finding this case indistinguishable from Antelope , we conclude that Broncheau has not been denied equal protection of the law because of his status as an Indian.

We have considered the other issues raised by Broncheau and find them to be without merit.

AFFIRMED.

. The Government concedes that Broncheau’s guilty plea does not foreclose appellate review of his constitutional challenges to § 1153, see e. g., Haynes v. United States, 390 U.S. 85 , 87 n. 2, 88 S.Ct. 722 , 19 L.Ed.2d 923 (1968), but contends that Broncheau’s attack on the adequacy of the indictment is not subject to review by this court. In general, a plea of guilty waives all nonjurisdictional defects, but does not bar appeal of claims that the applicable statute is unconstitutional or that the indictment fails to state an offense. See C. Wright, 1 Federal Practice and Procedure § 175 at 379-80 (1969); United States v. Gotches, 547 F.2d 80, 82 (8th Cir. 1977); Mercado v. Rockefeller, 502 F.2d 666, 672 (2d Cir. 1974), cert. denied, 420 U.S. 925 , 95 S.Ct. 1120 , 43 L.Ed.2d 394 (1975); Coleman v. Burnett, 155 U.S.App.D.C. 302, 309 , 477 F.2d 1187, 1194 (1973); United States v. Sepe, 474 F.2d 784, 788 (5th Cir. 1973) (quoting United States v. Cox, 464 F.2d 937, 941 (6th Cir. 1972), aff'd, 486 F.2d 1044 (5th Cir. 1973) (en banc). If Broncheau’s claim were correct, the indictment would fail to state an offense against the United States and the district court would be deprived of jurisdiction. Because the defect complained of is jurisdictional, Broncheau’s claim is reviewable. . § 1152 provides: “Except as otherwise expressly provided by law, the general laws of the United States as to the punishment of offenses committed in any place within the sole and exclusive jurisdiction of the United States, except the District of Columbia, shall extend to the Indian country. “This section shall not extend to offenses committed by one Indian against the person or property of another Indian, nor to any Indian committing any offense in the Indian country who has been punished by the local law of the tribe, or to any case where, by treaty stipulations, the exclusive jurisdiction over such offenses is or may be secured to the Indian tribes respectively.” . A congressional task force has suggested that it would be more appropriate to find that tribal courts have concurrent jurisdiction. See American Indian Policy Review Commission Task Force Four, Report on Federal, State and Tribal Jurisdiction 37 (Comm. Print. 1976). But the Supreme Court has recently suggested that “[t]he issue of exclusive jurisdiction over major crimes was mooted for all practical purposes by the passage of the Indian Civil Rights Act of 1968 . . . .” Oliphant v. Suquamish Indian Tribe, 435 U.S. at 203-04 n. 14, 98 S.Ct. at 1018. . Broncheau was charged with committing an assault resulting in serious bodily injury of a non-Indian within Indian country. As a result of the Indian Crimes Act of 1976, this crime is defined and punished in accordance with federal law. Thus, under §§ 1153 and 113(f), Broncheau was subject to a fine of not more than $10,000 or imprisonment for not more than 10 years or both. However, if Broncheau had committed the same offense in the same locale but had been a non-Indian, he would have been subject to state jurisdiction and state law pursuant to United States v. McBratney, 104 U.S. 621 , 26 L.Ed. 869 (1881), and its progeny. In that case, Broncheau would have been subject to punishment by a fine of not less than $100 or by imprisonment in the county jail for not less than 3 months nor more than one year, or by imprisonment in the state penitentiary for a period not to exceed 3 years or by a fine not exceeding $1000. Idaho Code § 18-912 (1978 Supp.) . As discussed in note 4 supra, the disparity arises because of the decision in United States v. McBratney, 104 U.S. 621 , 26 L.Ed. 869 (1881). The Supreme Court, Congress and various commentators have since noted that the Supreme Court reached its conclusion in McBratney despite the broad language of § 1152, which would appear to cover the commission of offenses by non-Indians against non-Indians in Indian country. See United States v. Wheeler, 435 U.S. at 324 n. 21, 98 S.Ct. 1079 ; H.R.Rep.No.1038, 94th Cong., 2d Sess., reprinted in [1976] U.S.Code Cong. & Admin.News, p. 1126; D. Getches, D. Rosenfelt, C. Wilkinson, Federal Indian Law, p. 366 (1979); American Indian Policy Review Commission, Task Force Four, Report on Federal, State and Tribal Jurisdictions 39 (Comm. Print. 1976). The inconsistency, such as it is, could be eliminated only if the Supreme Court overrules McBratney and succeeding cases or if Congress so directs. . Under 18 U.S.C. § 1152 , an Indian or non-Indian committing the same offense as Broncheau, but on a federal enclave instead of in Indian country, would be subject to federal law. . The constitutional defect found in these cases was cured by the Indian Crimes Act of 1976.