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	onn0 4401	The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
|- On =/= (/)
Theorem	onm 4402	The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
|- E. x x e. On
Theorem	suceq 4403	Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A = B -> suc A = suc B )
Theorem	elsuci 4404	Membership in a successor. This one-way implication does not require that either A or B be sets. (Contributed by NM, 6-Jun-1994.)
|- ( A e. suc B -> ( A e. B \/ A = B ) )
Theorem	elsucg 4405	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
|- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
Theorem	elsuc2g 4406	Variant of membership in a successor, requiring that B rather than A be a set. (Contributed by NM, 28-Oct-2003.)
|- ( B e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
Theorem	elsuc 4407	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
|- A e. _V   =>    |- ( A e. suc B <-> ( A e. B \/ A = B ) )
Theorem	elsuc2 4408	Membership in a successor. (Contributed by NM, 15-Sep-2003.)
|- A e. _V   =>    |- ( B e. suc A <-> ( B e. A \/ B = A ) )
Theorem	nfsuc 4409	Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
|- F/_ x A   =>    |- F/_ x suc A
Theorem	elelsuc 4410	Membership in a successor. (Contributed by NM, 20-Jun-1998.)
|- ( A e. B -> A e. suc B )
Theorem	sucel 4411*	Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
|- ( suc A e. B <-> E. x e. B A. y ( y e. x <-> ( y e. A \/ y = A ) ) )
Theorem	suc0 4412	The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
|- suc (/) = { (/) }
Theorem	sucprc 4413	A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
|- ( -. A e. _V -> suc A = A )
Theorem	unisuc 4414	A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   =>    |- ( Tr A <-> U. suc A = A )
Theorem	unisucg 4415	A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by Jim Kingdon, 18-Aug-2019.)
|- ( A e. V -> ( Tr A <-> U. suc A = A ) )
Theorem	sssucid 4416	A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)
|- A C_ suc A
Theorem	sucidg 4417	Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
|- ( A e. V -> A e. suc A )
Theorem	sucid 4418	A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
|- A e. _V   =>    |- A e. suc A
Theorem	nsuceq0g 4419	No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
|- ( A e. V -> suc A =/= (/) )
Theorem	eqelsuc 4420	A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
|- A e. _V   =>    |- ( A = B -> A e. suc B )
Theorem	iunsuc 4421*	Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
|- A e. _V   &    |- ( x = A -> B = C )   =>    |- U_ x e. suc A B = ( U_ x e. A B u. C )
Theorem	suctr 4422	The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
|- ( Tr A -> Tr suc A )
Theorem	trsuc 4423	A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- ( ( Tr A /\ suc B e. A ) -> B e. A )
Theorem	trsucss 4424	A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
|- ( Tr A -> ( B e. suc A -> B C_ A ) )
Theorem	sucssel 4425	A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
|- ( A e. V -> ( suc A C_ B -> A e. B ) )
Theorem	orduniss 4426	An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
|- ( Ord A -> U. A C_ A )
Theorem	onordi 4427	An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- Ord A
Theorem	ontrci 4428	An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- Tr A
Theorem	oneli 4429	A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- ( B e. A -> B e. On )
Theorem	onelssi 4430	A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
|- A e. On   =>    |- ( B e. A -> B C_ A )
Theorem	onelini 4431	An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- ( B e. A -> B = ( B i^i A ) )
Theorem	oneluni 4432	An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
|- A e. On   =>    |- ( B e. A -> ( A u. B ) = A )
Theorem	onunisuci 4433	An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
|- A e. On   =>    |- U. suc A = A
2.4  IZF Set Theory - add the Axiom of Union
2.4.1  Introduce the Axiom of Union
Axiom	ax-un 4434*	Axiom of Union. An axiom of Intuitionistic Zermelo-Fraenkel set theory. It states that a set y exists that includes the union of a given set x i.e. the collection of all members of the members of x. The variant axun2 4436 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4437. A version using class notation is uniex 4438. This is Axiom 3 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4125), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 266). The union of a class df-uni 3811 should not be confused with the union of two classes df-un 3134. Their relationship is shown in unipr 3824. (Contributed by NM, 23-Dec-1993.)
|- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
Theorem	zfun 4435*	Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
|- E. x A. y ( E. x ( y e. x /\ x e. z ) -> y e. x )
Theorem	axun2 4436*	A variant of the Axiom of Union ax-un 4434. For any set x, there exists a set y whose members are exactly the members of the members of x i.e. the union of x. Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
|- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )
Theorem	uniex2 4437*	The Axiom of Union using the standard abbreviation for union. Given any set x, its union y exists. (Contributed by NM, 4-Jun-2006.)
|- E. y y = U. x
Theorem	uniex 4438	The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2744), then the union of A is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.)
|- A e. _V   =>    |- U. A e. _V
Theorem	vuniex 4439	The union of a setvar is a set. (Contributed by BJ, 3-May-2021.)
|- U. x e. _V
Theorem	uniexg 4440	The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent A e. V instead of A e. _V to make the theorem more general and thus shorten some proofs; obviously the universal class constant _V is one possible substitution for class variable V. (Contributed by NM, 25-Nov-1994.)
|- ( A e. V -> U. A e. _V )
Theorem	uniexd 4441	Deduction version of the ZF Axiom of Union in class notation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
|- ( ph -> A e. V )   =>    |- ( ph -> U. A e. _V )
Theorem	unex 4442	The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)
|- A e. _V   &    |- B e. _V   =>    |- ( A u. B ) e. _V
Theorem	unexb 4443	Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
|- ( ( A e. _V /\ B e. _V ) <-> ( A u. B ) e. _V )
Theorem	unexg 4444	A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)
|- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V )
Theorem	tpexg 4445	An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)
|- ( ( A e. U /\ B e. V /\ C e. W ) -> { A , B , C } e. _V )
Theorem	unisn3 4446*	Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
|- ( A e. B -> U. { x e. B | x = A } = A )
Theorem	abnexg 4447*	Sufficient condition for a class abstraction to be a proper class. The class F can be thought of as an expression in x and the abstraction appearing in the statement as the class of values F as x varies through A. Assuming the antecedents, if that class is a set, then so is the "domain" A. The converse holds without antecedent, see abrexexg 6119. Note that the second antecedent A. x e. A x e. F cannot be translated to A C_ F since F may depend on x. In applications, one may take F = { x } or F = ~P x (see snnex 4449 and pwnex 4450 respectively, proved from abnex 4448, which is a consequence of abnexg 4447 with A = _V). (Contributed by BJ, 2-Dec-2021.)
|- ( A. x e. A ( F e. V /\ x e. F ) -> ( { y | E. x e. A y = F } e. W -> A e. _V ) )
Theorem	abnex 4448*	Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 4449 and pwnex 4450. See the comment of abnexg 4447. (Contributed by BJ, 2-May-2021.)
|- ( A. x ( F e. V /\ x e. F ) -> -. { y | E. x y = F } e. _V )
Theorem	snnex 4449*	The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.)
|- { x | E. y x = { y } } e/ _V
Theorem	pwnex 4450*	The class of all power sets is a proper class. See also snnex 4449. (Contributed by BJ, 2-May-2021.)
|- { x | E. y x = ~P y } e/ _V
Theorem	opeluu 4451	Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)
|- A e. _V   &    |- B e. _V   =>    |- ( <. A , B >. e. C -> ( A e. U. U. C /\ B e. U. U. C ) )
Theorem	uniuni 4452*	Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)
|- U. U. A = U. { x | E. y ( x = U. y /\ y e. A ) }
Theorem	eusv1 4453*	Two ways to express single-valuedness of a class expression A ( x ). (Contributed by NM, 14-Oct-2010.)
|- ( E! y A. x y = A <-> E. y A. x y = A )
Theorem	eusvnf 4454*	Even if x is free in A, it is effectively bound when A ( x ) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- ( E! y A. x y = A -> F/_ x A )
Theorem	eusvnfb 4455*	Two ways to say that A ( x ) is a set expression that does not depend on x. (Contributed by Mario Carneiro, 18-Nov-2016.)
|- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )
Theorem	eusv2i 4456*	Two ways to express single-valuedness of a class expression A ( x ). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)
|- ( E! y A. x y = A -> E! y E. x y = A )
Theorem	eusv2nf 4457*	Two ways to express single-valuedness of a class expression A ( x ). (Contributed by Mario Carneiro, 18-Nov-2016.)
|- A e. _V   =>    |- ( E! y E. x y = A <-> F/_ x A )
Theorem	eusv2 4458*	Two ways to express single-valuedness of a class expression A ( x ). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
|- A e. _V   =>    |- ( E! y E. x y = A <-> E! y A. x y = A )
Theorem	reusv1 4459*	Two ways to express single-valuedness of a class expression C ( y ). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
|- ( E. y e. B ph -> ( E! x e. A A. y e. B ( ph -> x = C ) <-> E. x e. A A. y e. B ( ph -> x = C ) ) )
Theorem	reusv3i 4460*	Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
|- ( y = z -> ( ph <-> ps ) )   &    |- ( y = z -> C = D )   =>    |- ( E. x e. A A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
Theorem	reusv3 4461*	Two ways to express single-valuedness of a class expression C ( y ). See reusv1 4459 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
|- ( y = z -> ( ph <-> ps ) )   &    |- ( y = z -> C = D )   =>    |- ( E. y e. B ( ph /\ C e. A ) -> ( A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) <-> E. x e. A A. y e. B ( ph -> x = C ) ) )
Theorem	alxfr 4462*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 18-Feb-2007.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A. y A e. B /\ A. x E. y x = A ) -> ( A. x ph <-> A. y ps ) )
Theorem	ralxfrd 4463*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
|- ( ( ph /\ y e. C ) -> A e. B )   &    |- ( ( ph /\ x e. B ) -> E. y e. C x = A )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )
Theorem	rexxfrd 4464*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
|- ( ( ph /\ y e. C ) -> A e. B )   &    |- ( ( ph /\ x e. B ) -> E. y e. C x = A )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) )
Theorem	ralxfr2d 4465*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by Mario Carneiro, 20-Aug-2014.)
|- ( ( ph /\ y e. C ) -> A e. V )   &    |- ( ph -> ( x e. B <-> E. y e. C x = A ) )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) )
Theorem	rexxfr2d 4466*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
|- ( ( ph /\ y e. C ) -> A e. V )   &    |- ( ph -> ( x e. B <-> E. y e. C x = A ) )   &    |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) )
Theorem	ralxfr 4467*	Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
|- ( y e. C -> A e. B )   &    |- ( x e. B -> E. y e. C x = A )   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x e. B ph <-> A. y e. C ps )
Theorem	ralxfrALT 4468*	Transfer universal quantification from a variable x to another variable y contained in expression A. This proof does not use ralxfrd 4463. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( y e. C -> A e. B )   &    |- ( x e. B -> E. y e. C x = A )   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x e. B ph <-> A. y e. C ps )
Theorem	rexxfr 4469*	Transfer existence from a variable x to another variable y contained in expression A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
|- ( y e. C -> A e. B )   &    |- ( x e. B -> E. y e. C x = A )   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( E. x e. B ph <-> E. y e. C ps )
Theorem	rabxfrd 4470*	Class builder membership after substituting an expression A (containing y) for x in the class expression ch. (Contributed by NM, 16-Jan-2012.)
|- F/_ y B   &    |- F/_ y C   &    |- ( ( ph /\ y e. D ) -> A e. D )   &    |- ( x = A -> ( ps <-> ch ) )   &    |- ( y = B -> A = C )   =>    |- ( ( ph /\ B e. D ) -> ( C e. { x e. D | ps } <-> B e. { y e. D | ch } ) )
Theorem	rabxfr 4471*	Class builder membership after substituting an expression A (containing y) for x in the class expression ph. (Contributed by NM, 10-Jun-2005.)
|- F/_ y B   &    |- F/_ y C   &    |- ( y e. D -> A e. D )   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> A = C )   =>    |- ( B e. D -> ( C e. { x e. D | ph } <-> B e. { y e. D | ps } ) )
Theorem	reuhypd 4472*	A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.)
|- ( ( ph /\ x e. C ) -> B e. C )   &    |- ( ( ph /\ x e. C /\ y e. C ) -> ( x = A <-> y = B ) )   =>    |- ( ( ph /\ x e. C ) -> E! y e. C x = A )
Theorem	reuhyp 4473*	A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.)
|- ( x e. C -> B e. C )   &    |- ( ( x e. C /\ y e. C ) -> ( x = A <-> y = B ) )   =>    |- ( x e. C -> E! y e. C x = A )
Theorem	uniexb 4474	The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.)
|- ( A e. _V <-> U. A e. _V )
Theorem	pwexb 4475	The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)
|- ( A e. _V <-> ~P A e. _V )
Theorem	elpwpwel 4476	A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.)
|- ( A e. ~P ~P B <-> U. A e. ~P B )
Theorem	univ 4477	The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
|- U. _V = _V
Theorem	eldifpw 4478	Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
|- C e. _V   =>    |- ( ( A e. ~P B /\ -. C C_ B ) -> ( A u. C ) e. ( ~P ( B u. C ) \ ~P B ) )
Theorem	op1stb 4479	Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.)
|- A e. _V   &    |- B e. _V   =>    |- |^| |^| <. A , B >. = A
Theorem	op1stbg 4480	Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)
|- ( ( A e. V /\ B e. W ) -> |^| |^| <. A , B >. = A )
Theorem	iunpw 4481*	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 )
Theorem	ifelpwung 4482	Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.)
|- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. ~P ( A u. B ) )
Theorem	ifelpwund 4483	Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> if ( ps , A , B ) e. ~P ( A u. B ) )
Theorem	ifelpwun 4484	Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.)
|- A e. _V   &    |- B e. _V   =>    |- if ( ph , A , B ) e. ~P ( A u. B )
Theorem	ifexd 4485	Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> if ( ps , A , B ) e. _V )
2.4.2  Ordinals (continued)
Theorem	ordon 4486	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 4487	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 4488	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 4489	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 4490	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 4491	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 4492	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 4493	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 4494	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 4495	The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
|- ( Ord A -> Ord U. A )
Theorem	bm2.5ii 4496*	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 4497	A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
|- ( A e. _V <-> suc A e. _V )
Theorem	sucexg 4498	The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
|- ( A e. V -> suc A e. _V )
Theorem	sucex 4499	The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   =>    |- suc A e. _V
Theorem	ordsucim 4500	The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
|- ( Ord A -> Ord suc A )