P. 152 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15101-15200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-vprc 15101	vprc 4150 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. _V
Theorem	bj-nvel 15102	nvel 4151 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. A
Theorem	bj-vnex 15103	vnex 4149 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x x = _V
Theorem	bdinex1 15104	Bounded version of inex1 4152. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( A i^i B ) e. _V
Theorem	bdinex2 15105	Bounded version of inex2 4153. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( B i^i A ) e. _V
Theorem	bdinex1g 15106	Bounded version of inex1g 4154. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   =>    |- ( A e. V -> ( A i^i B ) e. _V )
Theorem	bdssex 15107	Bounded version of ssex 4155. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   =>    |- ( A C_ B -> A e. _V )
Theorem	bdssexi 15108	Bounded version of ssexi 4156. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   &    |- A C_ B   =>    |- A e. _V
Theorem	bdssexg 15109	Bounded version of ssexg 4157. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( ( A C_ B /\ B e. C ) -> A e. _V )
Theorem	bdssexd 15110	Bounded version of ssexd 4158. (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 15111*	Bounded version of rabexg 4161. (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 15112	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 15113	intexr 4168 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^| A e. _V -> A =/= (/) )
Theorem	bj-intnexr 15114	intnexr 4169 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^| A = _V -> -. |^| A e. _V )
Theorem	bj-zfpair2 15115	Proof of zfpair2 4228 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- { x , y } e. _V
Theorem	bj-prexg 15116	Proof of prexg 4229 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 15117	snexg 4202 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- ( A e. V -> { A } e. _V )
Theorem	bj-snex 15118	snex 4203 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- { A } e. _V
Theorem	bj-sels 15119*	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 15120*	axun2 4453 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 15121*	uniex2 4454 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
|- E. y y = U. x
Theorem	bj-uniex 15122	uniex 4455 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- U. A e. _V
Theorem	bj-uniexg 15123	uniexg 4457 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> U. A e. _V )
Theorem	bj-unex 15124	unex 4459 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 15125	Bounded version of unexb 4460. (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 15126	unexg 4461 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 15127	sucexg 4515 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> suc A e. _V )
Theorem	bj-sucex 15128	sucex 4516 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- suc A e. _V
13.2.9.1  Delta_0-classical logic
Axiom	ax-bj-d0cl 15129	Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
|- BOUNDED ph   =>    |- DECID ph
Theorem	bj-d0clsepcl 15130	Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
|- DECID ph
13.2.9.2  Inductive classes and the class of natural number ordinals
Syntax	wind 15131	Syntax for inductive classes.
wff Ind A
Definition	df-bj-ind 15132*	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 15133	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 15134	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
|- ( A = B -> (Ind A <-> Ind B ) )
Theorem	bj-bdind 15135	Boundedness of the formula "the setvar x is an inductive class". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED Ind x
Theorem	bj-indint 15136*	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 15137*	If A is inductive and B is "inductive in A", 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 15138	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 15139	om is an inductive class. (Contributed by BJ, 30-Nov-2019.)
|- Ind om
Theorem	bj-omssind 15140	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 15141*	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 15142*	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 15143*	Two formulations of the axiom of infinity (see ax-infvn 15146 and bj-omex 15147) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( om e. _V <-> E. x (Ind x /\ A. y (Ind y -> x C_ y ) ) )
13.2.9.3  The first three Peano postulates The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4611 and peano3 4613 already show this. In this section, we prove bj-peano2 15144 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.
Theorem	bj-peano2 15144	Constructive proof of peano2 4612. Temporary note: another possibility is to simply replace sucexg 4515 with bj-sucexg 15127 in the proof of peano2 4612. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> suc A e. om )
Theorem	peano5set 15145*	Version of peano5 4615 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 ) )
13.2.10  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.
13.2.10.1  The set of natural number ordinals In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 15146) and deduce that the class om of natural number ordinals is a set (bj-omex 15147).
Axiom	ax-infvn 15146*	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 4605) from which one then proves, using full separation, that the wanted set exists (omex 4610). "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 15147	Proof of omex 4610 from ax-infvn 15146. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
|- om e. _V
13.2.10.2  Peano's fifth postulate In this section, we give constructive proofs of two versions of Peano's fifth postulate.
Theorem	bdpeano5 15148*	Bounded version of peano5 4615. (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 15149*	Version of peano5 4615 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 )
13.2.10.3  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 15150*	Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4616 for a nonconstructive proof of the general case. See bdfind 15151 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 15151*	Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4616 for a nonconstructive proof of the general case. See findset 15150 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 15152*	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 4617 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4617, finds2 4618, finds1 4619. (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 15153*	Version of bj-bdfindis 15152 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 15152 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 15154	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 15152 for explanations. From this version, it is easy to prove the bounded version of findes 4620. (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 15155*	Constructive proof of a variant of nn0suc 4621. For a constructive proof of nn0suc 4621, see bj-nn0suc 15169. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )
Theorem	bj-nntrans 15156	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 15157	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 15158	A natural number does not belong to itself. Version of elirr 4558 for natural numbers, which does not require ax-setind 4554. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> -. A e. A )
Theorem	bj-nnen2lp 15159	A version of en2lp 4571 for natural numbers, which does not require ax-setind 4554. Note: using this theorem and bj-nnelirr 15158, one can remove dependency on ax-setind 4554 from nntri2 6518 and nndcel 6524; 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 15160	Remove from peano4 4614 dependency on ax-setind 4554. 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 15161	The set om is transitive. A natural number is included in om. Constructive proof of elnn 4623. 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 15162	The set om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Tr om
Theorem	bj-nnord 15163	A natural number is an ordinal class. Constructive proof of nnord 4629. Can also be proved from bj-nnelon 15164 if the latter is proved from bj-omssonALT 15168. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> Ord A )
Theorem	bj-nnelon 15164	A natural number is an ordinal. Constructive proof of nnon 4627. Can also be proved from bj-omssonALT 15168. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> A e. On )
Theorem	bj-omord 15165	The set om is an ordinal class. Constructive proof of ordom 4624. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Ord om
Theorem	bj-omelon 15166	The set om is an ordinal. Constructive proof of omelon 4626. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- om e. On
Theorem	bj-omsson 15167	Constructive proof of omsson 4630. See also bj-omssonALT 15168. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
|- om C_ On
Theorem	bj-omssonALT 15168	Alternate proof of bj-omsson 15167. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
|- om C_ On
Theorem	bj-nn0suc 15169*	Proof of (biconditional form of) nn0suc 4621 from the core axioms of CZF. See also bj-nn0sucALT 15183. As a characterization of the elements of om, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om <-> ( A = (/) \/ E. x e. om A = suc x ) )
13.2.11  CZF: Set induction In this section, we add the axiom of set induction to the core axioms of CZF.
13.2.11.1  Set induction In this section, we prove some variants of the axiom of set induction.
Theorem	setindft 15170*	Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
|- ( A. x F/ y ph -> ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph ) )
Theorem	setindf 15171*	Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
|- F/ y ph   =>    |- ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph )
Theorem	setindis 15172*	Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
|- F/ x ps   &    |- F/ x ch   &    |- F/ y ph   &    |- F/ y ps   &    |- ( x = z -> ( ph -> ps ) )   &    |- ( x = y -> ( ch -> ph ) )   =>    |- ( A. y ( A. z e. y ps -> ch ) -> A. x ph )
Axiom	ax-bdsetind 15173*	Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
|- BOUNDED ph   =>    |- ( A. a ( A. y e. a [ y / a ] ph -> ph ) -> A. a ph )
Theorem	bdsetindis 15174*	Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- F/ x ps   &    |- F/ x ch   &    |- F/ y ph   &    |- F/ y ps   &    |- ( x = z -> ( ph -> ps ) )   &    |- ( x = y -> ( ch -> ph ) )   =>    |- ( A. y ( A. z e. y ps -> ch ) -> A. x ph )
Theorem	bj-inf2vnlem1 15175*	Lemma for bj-inf2vn 15179. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> Ind A )
Theorem	bj-inf2vnlem2 15176*	Lemma for bj-inf2vnlem3 15177 and bj-inf2vnlem4 15178. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A. u ( A. t e. u ( t e. A -> t e. Z ) -> ( u e. A -> u e. Z ) ) ) )
Theorem	bj-inf2vnlem3 15177*	Lemma for bj-inf2vn 15179. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- BOUNDED Z   =>    |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )
Theorem	bj-inf2vnlem4 15178*	Lemma for bj-inf2vn2 15180. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )
Theorem	bj-inf2vn 15179*	A sufficient condition for om to be a set. See bj-inf2vn2 15180 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( A e. V -> ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A = om ) )
Theorem	bj-inf2vn2 15180*	A sufficient condition for om to be a set; unbounded version of bj-inf2vn 15179. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- ( A e. V -> ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A = om ) )
Axiom	ax-inf2 15181*	Another axiom of infinity in a constructive setting (see ax-infvn 15146). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
|- E. a A. x ( x e. a <-> ( x = (/) \/ E. y e. a x = suc y ) )
Theorem	bj-omex2 15182	Using bounded set induction and the strong axiom of infinity, om is a set, that is, we recover ax-infvn 15146 (see bj-2inf 15143 for the equivalence of the latter with bj-omex 15147). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- om e. _V
Theorem	bj-nn0sucALT 15183*	Alternate proof of bj-nn0suc 15169, also constructive but from ax-inf2 15181, hence requiring ax-bdsetind 15173. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A e. om <-> ( A = (/) \/ E. x e. om A = suc x ) )
13.2.11.2  Full induction In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.
Theorem	bj-findis 15184*	Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 15152 for a bounded version not requiring ax-setind 4554. See finds 4617 for a proof in IZF. From this version, it is easy to prove of finds 4617, finds2 4618, finds1 4619. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.)
|- 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-findisg 15185*	Version of bj-findis 15184 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 15184 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- 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-findes 15186	Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 15184 for explanations. From this version, it is easy to prove findes 4620. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )
13.2.12  CZF: Strong collection In this section, we state the axiom scheme of strong collection, which is part of CZF set theory.
Axiom	ax-strcoll 15187*	Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that ph represents a multivalued function on a, or equivalently a collection of nonempty classes indexed by a, and the axiom asserts the existence of a set b which "collects" at least one element in the image of each x e. a and which is made only of such elements. That second conjunct is what makes it "strong", compared to the axiom scheme of collection ax-coll 4133. (Contributed by BJ, 5-Oct-2019.)
|- A. a ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
Theorem	strcoll2 15188*	Version of ax-strcoll 15187 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
|- ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
Theorem	strcollnft 15189*	Closed form of strcollnf 15190. (Contributed by BJ, 21-Oct-2019.)
|- ( A. x A. y F/ b ph -> ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) ) )
Theorem	strcollnf 15190*	Version of ax-strcoll 15187 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 15188 with the disjoint variable condition on b , ph replaced with a nonfreeness hypothesis. This proof aims to demonstrate a standard technique, but strcoll2 15188 will generally suffice: since the theorem asserts the existence of a set b, supposing that that setvar does not occur in the already defined ph is not a big constraint. (Contributed by BJ, 21-Oct-2019.)
|- F/ b ph   =>    |- ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
Theorem	strcollnfALT 15191*	Alternate proof of strcollnf 15190, not using strcollnft 15189. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ b ph   =>    |- ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
13.2.13  CZF: Subset collection In this section, we state the axiom scheme of subset collection, which is part of CZF set theory.
Axiom	ax-sscoll 15192*	Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that ph represents a multivalued function from a to b, or equivalently a collection of nonempty subsets of b indexed by a, and the consequent asserts the existence of a subset of c which "collects" at least one element in the image of each x e. a and which is made only of such elements. The axiom asserts the existence, for any sets a , b, of a set c such that that implication holds for any value of the parameter z of ph. (Contributed by BJ, 5-Oct-2019.)
|- A. a A. b E. c A. z ( A. x e. a E. y e. b ph -> E. d e. c ( A. x e. a E. y e. d ph /\ A. y e. d E. x e. a ph ) )
Theorem	sscoll2 15193*	Version of ax-sscoll 15192 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
|- E. c A. z ( A. x e. a E. y e. b ph -> E. d e. c ( A. x e. a E. y e. d ph /\ A. y e. d E. x e. a ph ) )
13.2.14  Real numbers
Axiom	ax-ddkcomp 15194	Axiom of Dedekind completeness for Dedekind real numbers: every inhabited upper-bounded located set of reals has a real upper bound. Ideally, this axiom should be "proved" as "axddkcomp" for the real numbers constructed from IZF, and then Axiom ax-ddkcomp 15194 should be used in place of construction specific results. In particular, axcaucvg 7928 should be proved from it. (Contributed by BJ, 24-Oct-2021.)
|- ( ( ( A C_ RR /\ E. x x e. A ) /\ E. x e. RR A. y e. A y < x /\ A. x e. RR A. y e. RR ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) ) -> E. x e. RR ( A. y e. A y <_ x /\ ( ( B e. R /\ A. y e. A y <_ B ) -> x <_ B ) ) )
13.3  Mathbox for Jim Kingdon
13.3.1  Propositional and predicate logic
Theorem	nnnotnotr 15195	Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 851, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.)
|- -. -. ( -. -. ph -> ph )
13.3.2  Natural numbers
Theorem	1dom1el 15196	If a set is dominated by one, then any two of its elements are equal. (Contributed by Jim Kingdon, 23-Apr-2025.)
|- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> B = C )
Theorem	ss1oel2o 15197	Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4216 which more directly illustrates the contrast with el2oss1o 6467. (Contributed by Jim Kingdon, 8-Aug-2022.)
|- (EXMID <-> A. x ( x C_ 1o -> x e. 2o ) )
Theorem	nnti 15198	Ordering on a natural number generates a tight apartness. (Contributed by Jim Kingdon, 7-Aug-2022.)
|- ( ph -> A e. om )   =>    |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u _E v /\ -. v _E u ) ) )
Theorem	012of 15199	Mapping zero and one between NN0 and om style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( `' G |` { 0 , 1 } ) : { 0 , 1 } --> 2o
Theorem	2o01f 15200	Mapping zero and one between om and NN0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( G |` 2o ) : 2o --> { 0 , 1 }