P. 46 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ordsucim 4501	The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
|- ( Ord A -> Ord suc A )
Theorem	onsuc 4502	The successor of an ordinal number is an ordinal number. Closed form of onsuci 4517. Forward implication of onsucb 4504. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
|- ( A e. On -> suc A e. On )
Theorem	ordsucg 4503	The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
|- ( A e. _V -> ( Ord A <-> Ord suc A ) )
Theorem	onsucb 4504	A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 4502. (Contributed by NM, 9-Sep-2003.)
|- ( A e. On <-> suc A e. On )
Theorem	ordsucss 4505	The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)
|- ( Ord B -> ( A e. B -> suc A C_ B ) )
Theorem	ordelsuc 4506	A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)
|- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )
Theorem	onsucssi 4507	A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)
|- A e. On   &    |- B e. On   =>    |- ( A e. B <-> suc A C_ B )
Theorem	onsucmin 4508*	The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)
|- ( A e. On -> suc A = |^| { x e. On | A e. x } )
Theorem	onsucelsucr 4509	Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4531. However, the converse does hold where B is a natural number, as seen at nnsucelsuc 6494. (Contributed by Jim Kingdon, 17-Jul-2019.)
|- ( B e. On -> ( suc A e. suc B -> A e. B ) )
Theorem	onsucsssucr 4510	The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4528. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
|- ( ( A e. On /\ Ord B ) -> ( suc A C_ suc B -> A C_ B ) )
Theorem	sucunielr 4511	Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4532. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- ( suc A e. B -> A e. U. B )
Theorem	unon 4512	The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
|- U. On = On
Theorem	onuniss2 4513*	The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- ( A e. On -> U. { x e. On | x C_ A } = A )
Theorem	limon 4514	The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
|- Lim On
Theorem	ordunisuc2r 4515*	An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.)
|- ( Ord A -> ( A. x e. A suc x e. A -> A = U. A ) )
Theorem	onssi 4516	An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
|- A e. On   =>    |- A C_ On
Theorem	onsuci 4517	The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4502 and onsucb 4504. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
|- A e. On   =>    |- suc A e. On
Theorem	onintonm 4518*	The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)
|- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. On )
Theorem	onintrab2im 4519	An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( E. x e. On ph -> |^| { x e. On | ph } e. On )
Theorem	ordtriexmidlem 4520	Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4522 or weak linearity in ordsoexmid 4563) with a proposition ph. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)
|- { x e. { (/) } | ph } e. On
Theorem	ordtriexmidlem2 4521*	Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4522 or weak linearity in ordsoexmid 4563) with a proposition ph. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.)
|- ( { x e. { (/) } | ph } = (/) -> -. ph )
Theorem	ordtriexmid 4522*	Ordinal trichotomy implies the law of the excluded middle (that is, decidability of an arbitrary proposition). This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". Also see exmidontri 7240 which is much the same theorem but biconditionalized and using the EXMID notation. (Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.)
|- A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x )   =>    |- ( ph \/ -. ph )
Theorem	ontriexmidim 4523*	Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4522. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> DECID ph )
Theorem	ordtri2orexmid 4524*	Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)
|- A. x e. On A. y e. On ( x e. y \/ y C_ x )   =>    |- ( ph \/ -. ph )
Theorem	2ordpr 4525	Version of 2on 6428 with the definition of 2o expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- Ord { (/) , { (/) } }
Theorem	ontr2exmid 4526*	An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
|- A. x e. On A. y A. z e. On ( ( x C_ y /\ y e. z ) -> x e. z )   =>    |- ( ph \/ -. ph )
Theorem	ordtri2or2exmidlem 4527*	A set which is 2o if ph or (/) if -. ph is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- { x e. { (/) , { (/) } } | ph } e. On
Theorem	onsucsssucexmid 4528*	The converse of onsucsssucr 4510 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
|- A. x e. On A. y e. On ( x C_ y -> suc x C_ suc y )   =>    |- ( ph \/ -. ph )
Theorem	onsucelsucexmidlem1 4529*	Lemma for onsucelsucexmid 4531. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
Theorem	onsucelsucexmidlem 4530*	Lemma for onsucelsucexmid 4531. The set { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } appears as A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5868), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4520. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. On
Theorem	onsucelsucexmid 4531*	The converse of onsucelsucr 4509 implies excluded middle. On the other hand, if y is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4509 does hold, as seen at nnsucelsuc 6494. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- A. x e. On A. y e. On ( x e. y -> suc x e. suc y )   =>    |- ( ph \/ -. ph )
Theorem	ordsucunielexmid 4532*	The converse of sucunielr 4511 (where B is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- A. x e. On A. y e. On ( x e. U. y -> suc x e. y )   =>    |- ( ph \/ -. ph )
2.5  IZF Set Theory - add the Axiom of Set Induction
2.5.1  The ZF Axiom of Foundation would imply Excluded Middle
Theorem	regexmidlemm 4533*	Lemma for regexmid 4536. A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- E. y y e. A
Theorem	regexmidlem1 4534*	Lemma for regexmid 4536. If A has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- ( E. y ( y e. A /\ A. z ( z e. y -> -. z e. A ) ) -> ( ph \/ -. ph ) )
Theorem	reg2exmidlema 4535*	Lemma for reg2exmid 4537. If A has a minimal element (expressed by C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- ( E. u e. A A. v e. A u C_ v -> ( ph \/ -. ph ) )
Theorem	regexmid 4536*	The axiom of foundation implies excluded middle. By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by e.). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4538. (Contributed by Jim Kingdon, 3-Sep-2019.)
|- ( E. y y e. x -> E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) )   =>    |- ( ph \/ -. ph )
Theorem	reg2exmid 4537*	If any inhabited set has a minimal element (when expressed by C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
|- A. z ( E. w w e. z -> E. x e. z A. y e. z x C_ y )   =>    |- ( ph \/ -. ph )
2.5.2  Introduce the Axiom of Set Induction
Axiom	ax-setind 4538*	Axiom of e.-Induction (also known as set induction). An axiom of Intuitionistic Zermelo-Fraenkel set theory. Axiom 9 of [Crosilla] p. "Axioms of CZF and IZF". This replaces the Axiom of Foundation (also called Regularity) from Zermelo-Fraenkel set theory. For more on axioms which might be adopted which are incompatible with this axiom (that is, Non-wellfounded Set Theory but in the absence of excluded middle), see Chapter 20 of [AczelRathjen], p. 183. (Contributed by Jim Kingdon, 19-Oct-2018.)
|- ( A. a ( A. y e. a [ y / a ] ph -> ph ) -> A. a ph )
Theorem	setindel 4539*	e.-Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.)
|- ( A. x ( A. y ( y e. x -> y e. S ) -> x e. S ) -> S = _V )
Theorem	setind 4540*	Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
|- ( A. x ( x C_ A -> x e. A ) -> A = _V )
Theorem	setind2 4541	Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
|- ( ~P A C_ A -> A = _V )
Theorem	elirr 4542	No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. The reason that this theorem is marked as discouraged is a bit subtle. If we wanted to reduce usage of ax-setind 4538, we could redefine Ord A (df-iord 4368) to also require _E Fr A (df-frind 4334) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4543 (which under that definition would presumably not need ax-setind 4538 to prove it). But since ordinals have not yet been defined that way, we cannot rely on the "don't add additional axiom use" feature of the minimizer to get theorems to use ordirr 4543. To encourage ordirr 4543 when possible, we mark this theorem as discouraged. (Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.)
|- -. A e. A
Theorem	ordirr 4543	Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The present proof requires ax-setind 4538. If in the definition of ordinals df-iord 4368, we also required that membership be well-founded on any ordinal (see df-frind 4334), then we could prove ordirr 4543 without ax-setind 4538. (Contributed by NM, 2-Jan-1994.)
|- ( Ord A -> -. A e. A )
Theorem	onirri 4544	An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- -. A e. A
Theorem	nordeq 4545	A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
|- ( ( Ord A /\ B e. A ) -> A =/= B )
Theorem	ordn2lp 4546	An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)
|- ( Ord A -> -. ( A e. B /\ B e. A ) )
Theorem	orddisj 4547	An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
|- ( Ord A -> ( A i^i { A } ) = (/) )
Theorem	orddif 4548	Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
|- ( Ord A -> A = ( suc A \ { A } ) )
Theorem	elirrv 4549	The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.)
|- -. x e. x
Theorem	sucprcreg 4550	A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.)
|- ( -. A e. _V <-> suc A = A )
Theorem	ruv 4551	The Russell class is equal to the universe _V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
|- { x | x e/ x } = _V
Theorem	ruALT 4552	Alternate proof of Russell's Paradox ru 2963, simplified using (indirectly) the Axiom of Set Induction ax-setind 4538. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
|- { x | x e/ x } e/ _V
Theorem	onprc 4553	No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4487), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.)
|- -. On e. _V
Theorem	sucon 4554	The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
|- suc On = On
Theorem	en2lp 4555	No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 27-Nov-2018.)
|- -. ( A e. B /\ B e. A )
Theorem	preleq 4556	Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )
Theorem	opthreg 4557	Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4538 (via the preleq 4556 step). See df-op 3603 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( { A , { A , B } } = { C , { C , D } } <-> ( A = C /\ B = D ) )
Theorem	suc11g 4558	The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
|- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) )
Theorem	suc11 4559	The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
|- ( ( A e. On /\ B e. On ) -> ( suc A = suc B <-> A = B ) )
Theorem	dtruex 4560*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4193 can also be summarized as "at least two sets exist", the difference is that dtruarb 4193 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific y, we can construct a set x which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- E. x -. x = y
Theorem	dtru 4561*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4560. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- -. A. x x = y
Theorem	eunex 4562	Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- ( E! x ph -> E. x -. ph )
Theorem	ordsoexmid 4563	Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.)
|- _E Or On   =>    |- ( ph \/ -. ph )
Theorem	ordsuc 4564	The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.)
|- ( Ord A <-> Ord suc A )
Theorem	onsucuni2 4565	A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( A e. On /\ A = suc B ) -> suc U. A = A )
Theorem	0elsucexmid 4566*	If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
|- A. x e. On (/) e. suc x   =>    |- ( ph \/ -. ph )
Theorem	nlimsucg 4567	A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A e. V -> -. Lim suc A )
Theorem	ordpwsucss 4568	The collection of ordinals in the power class of an ordinal is a superset of its successor. We can think of ( ~P A i^i On ) as another possible definition of successor, which would be equivalent to df-suc 4373 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A e. On then both U. suc A = A (onunisuci 4434) and U. { x e. On | x C_ A } = A (onuniss2 4513). Constructively ( ~P A i^i On ) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4571). (Contributed by Jim Kingdon, 21-Jul-2019.)
|- ( Ord A -> suc A C_ ( ~P A i^i On ) )
Theorem	onnmin 4569	No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.)
|- ( ( A C_ On /\ B e. A ) -> -. B e. |^| A )
Theorem	ssnel 4570	Relationship between subset and elementhood. In the context of ordinals this can be seen as an ordering law. (Contributed by Jim Kingdon, 22-Jul-2019.)
|- ( A C_ B -> -. B e. A )
Theorem	ordpwsucexmid 4571*	The subset in ordpwsucss 4568 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.)
|- A. x e. On suc x = ( ~P x i^i On )   =>    |- ( ph \/ -. ph )
Theorem	ordtri2or2exmid 4572*	Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- A. x e. On A. y e. On ( x C_ y \/ y C_ x )   =>    |- ( ph \/ -. ph )
Theorem	ontri2orexmidim 4573*	Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4572. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> DECID ph )
Theorem	onintexmid 4574*	If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
|- ( ( y C_ On /\ E. x x e. y ) -> |^| y e. y )   =>    |- ( ph \/ -. ph )
Theorem	zfregfr 4575	The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
|- _E Fr A
Theorem	ordfr 4576	Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
|- ( Ord A -> _E Fr A )
Theorem	ordwe 4577	Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
|- ( Ord A -> _E We A )
Theorem	wetriext 4578*	A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
|- ( ph -> R We A )   &    |- ( ph -> A e. V )   &    |- ( ph -> A. a e. A A. b e. A ( a R b \/ a = b \/ b R a ) )   &    |- ( ph -> B e. A )   &    |- ( ph -> C e. A )   &    |- ( ph -> A. z e. A ( z R B <-> z R C ) )   =>    |- ( ph -> B = C )
Theorem	wessep 4579	A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)
|- ( ( _E We A /\ B C_ A ) -> _E We B )
Theorem	reg3exmidlemwe 4580*	Lemma for reg3exmid 4581. Our counterexample A satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- _E We A
Theorem	reg3exmid 4581*	If any inhabited set satisfying df-wetr 4336 for _E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- ( ( _E We z /\ E. w w e. z ) -> E. x e. z A. y e. z x C_ y )   =>    |- ( ph \/ -. ph )
Theorem	dcextest 4582*	If it is decidable whether { x | ph } is a set, then -. ph is decidable (where x does not occur in ph). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition -. ph is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)
|- DECID { x | ph } e. _V   =>    |- DECID -. ph
2.5.3  Transfinite induction
Theorem	tfi 4583*	The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if A is a class of ordinal numbers with the property that every ordinal number included in A also belongs to A, then every ordinal number is in A. (Contributed by NM, 18-Feb-2004.)
|- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On )
Theorem	tfis 4584*	Transfinite Induction Schema. If all ordinal numbers less than a given number x have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.)
|- ( x e. On -> ( A. y e. x [ y / x ] ph -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis2f 4585*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis2 4586*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis3 4587*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = A -> ( ph <-> ch ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( A e. On -> ch )
Theorem	tfisi 4588*	A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.)
|- ( ph -> A e. V )   &    |- ( ph -> T e. On )   &    |- ( ( ph /\ ( R e. On /\ R C_ T ) /\ A. y ( S e. R -> ch ) ) -> ps )   &    |- ( x = y -> ( ps <-> ch ) )   &    |- ( x = A -> ( ps <-> th ) )   &    |- ( x = y -> R = S )   &    |- ( x = A -> R = T )   =>    |- ( ph -> th )
2.6  IZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity
Axiom	ax-iinf 4589*	Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.)
|- E. x ( (/) e. x /\ A. y ( y e. x -> suc y e. x ) )
Theorem	zfinf2 4590*	A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.)
|- E. x ( (/) e. x /\ A. y e. x suc y e. x )
2.6.2  The natural numbers
Syntax	com 4591	Extend class notation to include the class of natural numbers.
class om
Definition	df-iom 4592*	Define the class of natural numbers as the smallest inductive set, which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. Note: the natural numbers om are a subset of the ordinal numbers df-on 4370. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8922) with analogous properties and operations, but they will be different sets. We are unable to use the terms finite ordinal and natural number interchangeably, as shown at exmidonfin 7195. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4593 instead for naming consistency with set.mm. (New usage is discouraged.)
|- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
Theorem	dfom3 4593*	Alias for df-iom 4592. Use it instead of df-iom 4592 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.)
|- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
Theorem	omex 4594	The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.)
|- om e. _V
2.6.3  Peano's postulates
Theorem	peano1 4595	Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.)
|- (/) e. om
Theorem	peano2 4596	The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
|- ( A e. om -> suc A e. om )
Theorem	peano3 4597	The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
|- ( A e. om -> suc A =/= (/) )
Theorem	peano4 4598	Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.)
|- ( ( A e. om /\ B e. om ) -> ( suc A = suc B <-> A = B ) )
Theorem	peano5 4599*	The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as Theorem findes 4604. (Contributed by NM, 18-Feb-2004.)
|- ( ( (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A )
2.6.4  Finite induction (for finite ordinals)
Theorem	find 4600*	The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that A is a set of natural numbers, zero belongs to A, and given any member of A the member's successor also belongs to A. The conclusion is that every natural number is in A. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A )   =>    |- A = om