P. 170 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16901-17000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bdcin 16901	The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A i^i B )
Theorem	bdss 16902	The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED x C_ A
Theorem	bdcnul 16903	The empty class is bounded. See also bdcnulALT 16904. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED (/)
Theorem	bdcnulALT 16904	Alternate proof of bdcnul 16903. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 16882, or use the corresponding characterizations of its elements followed by bdelir 16885. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- BOUNDED (/)
Theorem	bdeq0 16905	Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)
|- BOUNDED x = (/)
Theorem	bj-bd0el 16906	Boundedness of the formula "the empty set belongs to the setvar x". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED (/) e. x
Theorem	bdcpw 16907	The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED ~P A
Theorem	bdcsn 16908	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x }
Theorem	bdcpr 16909	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y }
Theorem	bdctp 16910	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y , z }
Theorem	bdsnss 16911*	Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED { x } C_ A
Theorem	bdvsn 16912*	Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED x = { y }
Theorem	bdop 16913	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
|- BOUNDED <. x , y >.
Theorem	bdcuni 16914	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
|- BOUNDED U. x
Theorem	bdcint 16915	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED |^| x
Theorem	bdciun 16916*	The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED U_ x e. y A
Theorem	bdciin 16917*	The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED |^|_ x e. y A
Theorem	bdcsuc 16918	The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED suc x
Theorem	bdeqsuc 16919*	Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
|- BOUNDED x = suc y
Theorem	bj-bdsucel 16920	Boundedness of the formula "the successor of the setvar x belongs to the setvar y". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED suc x e. y
Theorem	bdcriota 16921*	A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
|- BOUNDED ph   &    |- E! x e. y ph   =>    |- BOUNDED ( iota_ x e. y ph )
14.3.9  CZF: Bounded separation In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.
Axiom	ax-bdsep 16922*	Axiom scheme of bounded (or restricted, or Δ0) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 4249. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- A. a E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsep1 16923*	Version of ax-bdsep 16922 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsep2 16924*	Version of ax-bdsep 16922 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 16923 when sufficient. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsepnft 16925*	Closed form of bdsepnf 16926. Version of ax-bdsep 16922 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 16923 when sufficient. (Contributed by BJ, 19-Oct-2019.)
|- BOUNDED ph   =>    |- ( A. x F/ b ph -> E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) )
Theorem	bdsepnf 16926*	Version of ax-bdsep 16922 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 16927. Use bdsep1 16923 when sufficient. (Contributed by BJ, 5-Oct-2019.)
|- F/ b ph   &    |- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsepnfALT 16927*	Alternate proof of bdsepnf 16926, not using bdsepnft 16925. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ b ph   &    |- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsepg 16928*	Version of sepg 4251 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
|- BOUNDED ph   =>    |- ( A e. V -> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) )
Theorem	bdbm1.3ii 16929*	Bounded version of bm1.3ii 4254. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- E. x A. y ( ph -> y e. x )   =>    |- E. x A. y ( y e. x <-> ph )
Theorem	bj-axemptylem 16930*	Lemma for bj-axempty 16931 and bj-axempty2 16932. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4259 instead. (New usage is discouraged.)
|- E. x A. y ( y e. x -> F. )
Theorem	bj-axempty 16931*	Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a nonempty universe. See axnul 4258. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4259 instead. (New usage is discouraged.)
|- E. x A. y e. x F.
Theorem	bj-axempty2 16932*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 16931. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4259 instead. (New usage is discouraged.)
|- E. x A. y -. y e. x
Theorem	bj-nalset 16933*	nalset 4263 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x A. y y e. x
Theorem	bj-vprc 16934	vprc 4265 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. _V
Theorem	bj-nvel 16935	nvel 4266 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. A
Theorem	bj-vnex 16936	vnex 4264 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x x = _V
Theorem	bdinex1 16937	Bounded version of inex1 4267. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( A i^i B ) e. _V
Theorem	bdinex2 16938	Bounded version of inex2 4268. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( B i^i A ) e. _V
Theorem	bdinex1g 16939	Bounded version of inex1g 4269. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   =>    |- ( A e. V -> ( A i^i B ) e. _V )
Theorem	bdssex 16940	Bounded version of ssex 4270. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   =>    |- ( A C_ B -> A e. _V )
Theorem	bdssexi 16941	Bounded version of ssexi 4271. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   &    |- A C_ B   =>    |- A e. _V
Theorem	bdssexg 16942	Bounded version of ssexg 4272. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( ( A C_ B /\ B e. C ) -> A e. _V )
Theorem	bdssexd 16943	Bounded version of ssexd 4273. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( ph -> B e. C )   &    |- ( ph -> A C_ B )   &    |- BOUNDED A   =>    |- ( ph -> A e. _V )
Theorem	bdrabexg 16944*	Bounded version of rabexg 4279. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- BOUNDED A   =>    |- ( A e. V -> { x e. A | ph } e. _V )
Theorem	bj-inex 16945	The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( ( A e. V /\ B e. W ) -> ( A i^i B ) e. _V )
Theorem	bj-intexr 16946	intexr 4286 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^| A e. _V -> A =/= (/) )
Theorem	bj-intnexr 16947	intnexr 4287 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^| A = _V -> -. |^| A e. _V )
Theorem	bj-zfpair2 16948	Proof of zfpair2 4347 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- { x , y } e. _V
Theorem	bj-prexg 16949	Proof of prexg 4349 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )
Theorem	bj-snexg 16950	snexg 4321 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- ( A e. V -> { A } e. _V )
Theorem	bj-snex 16951	snex 4322 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- { A } e. _V
Theorem	bj-sels 16952*	If a class is a set, then it is a member of a set. (Copied from set.mm.) (Contributed by BJ, 3-Apr-2019.)
|- ( A e. V -> E. x A e. x )
Theorem	bj-axun2 16953*	axun2 4580 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
|- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )
Theorem	bj-uniex2 16954*	uniex2 4581 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
|- E. y y = U. x
Theorem	bj-uniex 16955	uniex 4583 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- U. A e. _V
Theorem	bj-uniexg 16956	uniexg 4585 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> U. A e. _V )
Theorem	bj-unex 16957	unex 4587 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   &    |- B e. _V   =>    |- ( A u. B ) e. _V
Theorem	bdunexb 16958	Bounded version of unexb 4588. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V )
Theorem	bj-unexg 16959	unexg 4589 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V )
Theorem	bj-sucexg 16960	sucexg 4645 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> suc A e. _V )
Theorem	bj-sucex 16961	sucex 4646 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- suc A e. _V
14.3.9.1  Delta_0-classical logic
Axiom	ax-bj-d0cl 16962	Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.) New usage is discouraged since this statement is not intuitionnistic. (New usage is discouraged.)
|- BOUNDED ph   =>    |- DECID ph
Theorem	bj-d0clsepcl 16963	Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) New usage is discouraged since this statement is not intuitionnistic. (New usage is discouraged.)
|- DECID ph
14.3.9.2  Inductive classes and the class of natural number ordinals
Syntax	wind 16964	Syntax for inductive classes.
wff Ind A
Definition	df-bj-ind 16965*	Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
|- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
Theorem	bj-indsuc 16966	A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
|- (Ind A -> ( B e. A -> suc B e. A ) )
Theorem	bj-indeq 16967	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
|- ( A = B -> (Ind A <-> Ind B ) )
Theorem	bj-bdind 16968	Boundedness of the formula "the setvar x is an inductive class". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED Ind x
Theorem	bj-indint 16969*	The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
|- Ind |^| { x e. A | Ind x }
Theorem	bj-indind 16970*	If A is inductive and B is "inductive in A " (a condition weaker than "inductive"), then ( A i^i B ) is inductive. (Contributed by BJ, 25-Oct-2020.)
|- ( (Ind A /\ ( (/) e. B /\ A. x e. A ( x e. B -> suc x e. B ) ) ) -> Ind ( A i^i B ) )
Theorem	bj-dfom 16971	Alternate definition of om, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.)
|- om = |^| { x | Ind x }
Theorem	bj-omind 16972	om is an inductive class. (Contributed by BJ, 30-Nov-2019.)
|- Ind om
Theorem	bj-omssind 16973	om is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> (Ind A -> om C_ A ) )
Theorem	bj-ssom 16974*	A characterization of subclasses of om. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( A. x (Ind x -> A C_ x ) <-> A C_ om )
Theorem	bj-om 16975*	A set is equal to om if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> ( A = om <-> (Ind A /\ A. x (Ind x -> A C_ x ) ) ) )
Theorem	bj-2inf 16976*	Two formulations of the axiom of infinity (see ax-infvn 16979 and bj-omex 16980) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( om e. _V <-> E. x (Ind x /\ A. y (Ind y -> x C_ y ) ) )
14.3.9.3  The first three Peano postulates The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4741 and peano3 4743 already show this. In this section, we prove bj-peano2 16977 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.
Theorem	bj-peano2 16977	Constructive proof of peano2 4742. Temporary note: another possibility is to simply replace sucexg 4645 with bj-sucexg 16960 in the proof of peano2 4742. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> suc A e. om )
Theorem	peano5set 16978*	Version of peano5 4745 when om i^i A is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( ( om i^i A ) e. V -> ( ( (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A ) )
14.3.10  CZF: Infinity In the absence of full separation, the axiom of infinity has to be stated more precisely, as the existence of the smallest class containing the empty set and the successor of each of its elements.
14.3.10.1  The set of natural number ordinals In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 16979) and deduce that the class om of natural number ordinals is a set (bj-omex 16980).
Axiom	ax-infvn 16979*	Axiom of infinity in a constructive setting. This asserts the existence of the special set we want (the set of natural numbers), instead of the existence of a set with some properties (ax-iinf 4735) from which one then proves, using full separation, that the wanted set exists (omex 4740). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
|- E. x (Ind x /\ A. y (Ind y -> x C_ y ) )
Theorem	bj-omex 16980	Proof of omex 4740 from ax-infvn 16979. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
|- om e. _V
14.3.10.2  Peano's fifth postulate In this section, we give constructive proofs of two versions of Peano's fifth postulate.
Theorem	bdpeano5 16981*	Bounded version of peano5 4745. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( ( (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A )
Theorem	speano5 16982*	Version of peano5 4745 when A is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( ( A e. V /\ (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A )
14.3.10.3  Bounded induction and Peano's fourth postulate In this section, we prove various versions of bounded induction from the basic axioms of CZF (in particular, without the axiom of set induction). We also prove Peano's fourth postulate. Together with the results from the previous sections, this proves from the core axioms of CZF (with infinity) that the set of natural number ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers.
Theorem	findset 16983*	Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4746 for a nonconstructive proof of the general case. See bdfind 16984 for a proof when A is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A = om ) )
Theorem	bdfind 16984*	Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4746 for a nonconstructive proof of the general case. See findset 16983 for a proof when A is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A = om )
Theorem	bj-bdfindis 16985*	Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4747 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4747, finds2 4748, finds1 4749. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- F/ x ps   &    |- F/ x ch   &    |- F/ x th   &    |- ( x = (/) -> ( ps -> ph ) )   &    |- ( x = y -> ( ph -> ch ) )   &    |- ( x = suc y -> ( th -> ph ) )   =>    |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )
Theorem	bj-bdfindisg 16986*	Version of bj-bdfindis 16985 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 16985 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- F/ x ps   &    |- F/ x ch   &    |- F/ x th   &    |- ( x = (/) -> ( ps -> ph ) )   &    |- ( x = y -> ( ph -> ch ) )   &    |- ( x = suc y -> ( th -> ph ) )   &    |- F/_ x A   &    |- F/ x ta   &    |- ( x = A -> ( ph -> ta ) )   =>    |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> ( A e. om -> ta ) )
Theorem	bj-bdfindes 16987	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 16985 for explanations. From this version, it is easy to prove the bounded version of findes 4750. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   =>    |- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )
Theorem	bj-nn0suc0 16988*	Constructive proof of a variant of nn0suc 4751. For a constructive proof of nn0suc 4751, see bj-nn0suc 17002. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )
Theorem	bj-nntrans 16989	A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> ( B e. A -> B C_ A ) )
Theorem	bj-nntrans2 16990	A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> Tr A )
Theorem	bj-nnelirr 16991	A natural number does not belong to itself. Version of elirr 4688 for natural numbers, which does not require ax-setind 4684. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> -. A e. A )
Theorem	bj-nnen2lp 16992	A version of en2lp 4701 for natural numbers, which does not require ax-setind 4684. Note: using this theorem and bj-nnelirr 16991, one can remove dependency on ax-setind 4684 from nntri2 6767 and nndcel 6773; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)
|- ( ( A e. om /\ B e. om ) -> -. ( A e. B /\ B e. A ) )
Theorem	bj-peano4 16993	Remove from peano4 4744 dependency on ax-setind 4684. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)
|- ( ( A e. om /\ B e. om ) -> ( suc A = suc B <-> A = B ) )
Theorem	bj-omtrans 16994	The set om is transitive. A natural number is included in om. Constructive proof of elomssom 4752. The idea is to use bounded induction with the formula x C_ om. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with x C_ a and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- ( A e. om -> A C_ om )
Theorem	bj-omtrans2 16995	The set om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Tr om
Theorem	bj-nnord 16996	A natural number is an ordinal class. Constructive proof of nnord 4759. Can also be proved from bj-nnelon 16997 if the latter is proved from bj-omssonALT 17001. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> Ord A )
Theorem	bj-nnelon 16997	A natural number is an ordinal. Constructive proof of nnon 4757. Can also be proved from bj-omssonALT 17001. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> A e. On )
Theorem	bj-omord 16998	The set om is an ordinal class. Constructive proof of ordom 4754. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Ord om
Theorem	bj-omelon 16999	The set om is an ordinal. Constructive proof of omelon 4756. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- om e. On
Theorem	bj-omsson 17000	Constructive proof of omsson 4760. See also bj-omssonALT 17001. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
|- om C_ On