P. 45 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4401-4500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iunpw 4401*	An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
|- A e. _V   =>    |- ( E. x e. A x = U. A <-> ~P U. A = U_ x e. A ~P x )
2.4.2  Ordinals (continued)
Theorem	ordon 4402	The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
|- Ord On
Theorem	ssorduni 4403	The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- ( A C_ On -> Ord U. A )
Theorem	ssonuni 4404	The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)
|- ( A e. V -> ( A C_ On -> U. A e. On ) )
Theorem	ssonunii 4405	The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.)
|- A e. _V   =>    |- ( A C_ On -> U. A e. On )
Theorem	onun2 4406	The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)
|- ( ( A e. On /\ B e. On ) -> ( A u. B ) e. On )
Theorem	onun2i 4407	The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.)
|- A e. On   &    |- B e. On   =>    |- ( A u. B ) e. On
Theorem	ordsson 4408	Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.)
|- ( Ord A -> A C_ On )
Theorem	onss 4409	An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
|- ( A e. On -> A C_ On )
Theorem	onuni 4410	The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
|- ( A e. On -> U. A e. On )
Theorem	orduni 4411	The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
|- ( Ord A -> Ord U. A )
Theorem	bm2.5ii 4412*	Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
|- A e. _V   =>    |- ( A C_ On -> U. A = |^| { x e. On | A. y e. A y C_ x } )
Theorem	sucexb 4413	A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
|- ( A e. _V <-> suc A e. _V )
Theorem	sucexg 4414	The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
|- ( A e. V -> suc A e. _V )
Theorem	sucex 4415	The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   =>    |- suc A e. _V
Theorem	ordsucim 4416	The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
|- ( Ord A -> Ord suc A )
Theorem	suceloni 4417	The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
|- ( A e. On -> suc A e. On )
Theorem	ordsucg 4418	The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
|- ( A e. _V -> ( Ord A <-> Ord suc A ) )
Theorem	sucelon 4419	The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
|- ( A e. On <-> suc A e. On )
Theorem	ordsucss 4420	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 4421	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 4422	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 4423*	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 4424	Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4445. However, the converse does hold where B is a natural number, as seen at nnsucelsuc 6387. (Contributed by Jim Kingdon, 17-Jul-2019.)
|- ( B e. On -> ( suc A e. suc B -> A e. B ) )
Theorem	onsucsssucr 4425	The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4442. (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 4426	Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4446. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- ( suc A e. B -> A e. U. B )
Theorem	unon 4427	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 4428*	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 4429	The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
|- Lim On
Theorem	ordunisuc2r 4430*	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 4431	An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
|- A e. On   =>    |- A C_ On
Theorem	onsuci 4432	The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
|- A e. On   =>    |- suc A e. On
Theorem	onintonm 4433*	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 4434	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 4435	Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4437 or weak linearity in ordsoexmid 4477) 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 4436*	Lemma for decidability and ordinals. The set { x e. { (/) } | ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4437 or weak linearity in ordsoexmid 4477) 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 4437*	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". (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	ordtri2orexmid 4438*	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 4439	Version of 2on 6322 with the definition of 2o expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- Ord { (/) , { (/) } }
Theorem	ontr2exmid 4440*	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 4441*	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 4442*	The converse of onsucsssucr 4425 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 4443*	Lemma for onsucelsucexmid 4445. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }
Theorem	onsucelsucexmidlem 4444*	Lemma for onsucelsucexmid 4445. The set { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } appears as A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5765), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4435. (Contributed by Jim Kingdon, 2-Aug-2019.)
|- { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } e. On
Theorem	onsucelsucexmid 4445*	The converse of onsucelsucr 4424 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 4424 does hold, as seen at nnsucelsuc 6387. (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 4446*	The converse of sucunielr 4426 (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 4447*	Lemma for regexmid 4450. A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- E. y y e. A
Theorem	regexmidlem1 4448*	Lemma for regexmid 4450. 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 4449*	Lemma for reg2exmid 4451. 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 4450*	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 4452. (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 4451*	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 4452*	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 4453*	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 4454*	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 4455	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 4456	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 4452, we could redefine Ord A (df-iord 4288) to also require _E Fr A (df-frind 4254) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4457 (which under that definition would presumably not need ax-setind 4452 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 4457. To encourage ordirr 4457 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 4457	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 4452. If in the definition of ordinals df-iord 4288, we also required that membership be well-founded on any ordinal (see df-frind 4254), then we could prove ordirr 4457 without ax-setind 4452. (Contributed by NM, 2-Jan-1994.)
|- ( Ord A -> -. A e. A )
Theorem	onirri 4458	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 4459	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 4460	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 4461	An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
|- ( Ord A -> ( A i^i { A } ) = (/) )
Theorem	orddif 4462	Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
|- ( Ord A -> A = ( suc A \ { A } ) )
Theorem	elirrv 4463	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 4464	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 4465	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 4466	Alternate proof of Russell's Paradox ru 2908, simplified using (indirectly) the Axiom of Set Induction ax-setind 4452. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
|- { x | x e/ x } e/ _V
Theorem	onprc 4467	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 4402), 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 4468	The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
|- suc On = On
Theorem	en2lp 4469	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 4470	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 4471	Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4452 (via the preleq 4470 step). See df-op 3536 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 4472	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 4473	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 4474*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4115 can also be summarized as "at least two sets exist", the difference is that dtruarb 4115 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 4475*	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 4474. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- -. A. x x = y
Theorem	eunex 4476	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 4477	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 4478	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 4479	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 4480*	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 4481	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 4482	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 4293 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 4354) and U. { x e. On | x C_ A } = A (onuniss2 4428). Constructively ( ~P A i^i On ) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4485). (Contributed by Jim Kingdon, 21-Jul-2019.)
|- ( Ord A -> suc A C_ ( ~P A i^i On ) )
Theorem	onnmin 4483	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 4484	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 4485*	The subset in ordpwsucss 4482 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 4486*	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	onintexmid 4487*	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 4488	The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
|- _E Fr A
Theorem	ordfr 4489	Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
|- ( Ord A -> _E Fr A )
Theorem	ordwe 4490	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 4491*	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 4492	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 4493*	Lemma for reg3exmid 4494. Our counterexample A satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- _E We A
Theorem	reg3exmid 4494*	If any inhabited set satisfying df-wetr 4256 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 4495*	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 4496*	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 4497*	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 4498*	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 4499*	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 4500*	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 )