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Theorem List for Intuitionistic Logic Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonm 4201 The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
 |- 
 E. x  x  e. 
 On
 
Theoremsuceq 4202 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  =  B  ->  suc  A  =  suc  B )
 
Theoremelsuci 4203 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 ) )
 
Theoremelsucg 4204 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 ) ) )
 
Theoremelsuc2g 4205 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 ) ) )
 
Theoremelsuc 4206 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 ) )
 
Theoremelsuc2 4207 Membership in a successor. (Contributed by NM, 15-Sep-2003.)
 |-  A  e.  _V   =>    |-  ( B  e.  suc 
 A 
 <->  ( B  e.  A  \/  B  =  A ) )
 
Theoremnfsuc 4208 Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
 |-  F/_ x A   =>    |-  F/_ x  suc  A
 
Theoremelelsuc 4209 Membership in a successor. (Contributed by NM, 20-Jun-1998.)
 |-  ( A  e.  B  ->  A  e.  suc  B )
 
Theoremsucel 4210* 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 ) ) )
 
Theoremsuc0 4211 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
 |- 
 suc  (/)  =  { (/) }
 
Theoremsucprc 4212 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
 |-  ( -.  A  e.  _V 
 ->  suc  A  =  A )
 
Theoremunisuc 4213 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 )
 
Theoremunisucg 4214 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 ) )
 
Theoremsssucid 4215 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
 
Theoremsucidg 4216 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 )
 
Theoremsucid 4217 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
 
Theoremnsuceq0g 4218 No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
 |-  ( A  e.  V  ->  suc  A  =/=  (/) )
 
Theoremeqelsuc 4219 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 )
 
Theoremiunsuc 4220* 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 )
 
Theoremsuctr 4221 The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
 |-  ( Tr  A  ->  Tr 
 suc  A )
 
Theoremtrsuc 4222 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 )
 
Theoremtrsucss 4223 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 )
 )
 
Theoremsucssel 4224 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 ) )
 
Theoremorduniss 4225 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
 |-  ( Ord  A  ->  U. A  C_  A )
 
Theoremonordi 4226 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  On   =>    |-  Ord  A
 
Theoremontrci 4227 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  On   =>    |-  Tr  A
 
Theoremoneli 4228 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 )
 
Theoremonelssi 4229 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 )
 
Theoremonelini 4230 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 ) )
 
Theoremoneluni 4231 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 )
 
Theoremonunisuci 4232 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
 
Axiomax-un 4233* 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 4235 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4236. A version using class notation is uniex 4237.

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 3934), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 262).

The union of a class df-uni 3637 should not be confused with the union of two classes df-un 2992. Their relationship is shown in unipr 3650. (Contributed by NM, 23-Dec-1993.)

 |- 
 E. y A. z
 ( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
 
Theoremzfun 4234* 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 )
 
Theoremaxun2 4235* A variant of the Axiom of Union ax-un 4233. 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 ) )
 
Theoremuniex2 4236* 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
 
Theoremuniex 4237 The Axiom of Union in class notation. This says that if  A is a set i.e.  A  e.  _V (see isset 2619), 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
 
Theoremuniexg 4238 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 )
 
Theoremunex 4239 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
 
Theoremunexb 4240 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 )
 
Theoremunexg 4241 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 )
 
Theoremtpexg 4242 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 )
 
Theoremunisn3 4243* 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 )
 
Theoremsnnex 4244* 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
 
Theoremopeluu 4245 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 ) )
 
Theoremuniuni 4246* 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 ) }
 
Theoremeusv1 4247* 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 )
 
Theoremeusvnf 4248* 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 )
 
Theoremeusvnfb 4249* 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 )
 )
 
Theoremeusv2i 4250* 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 )
 
Theoremeusv2nf 4251* 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 )
 
Theoremeusv2 4252* 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 )
 
Theoremreusv1 4253* 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 ) ) )
 
Theoremreusv3i 4254* 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 ) )
 
Theoremreusv3 4255* Two ways to express single-valuedness of a class expression  C ( y ). See reusv1 4253 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 ) ) )
 
Theoremalxfr 4256* 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 ) )
 
Theoremralxfrd 4257* 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 ) )
 
Theoremrexxfrd 4258* 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 ) )
 
Theoremralxfr2d 4259* 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 ) )
 
Theoremrexxfr2d 4260* 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 ) )
 
Theoremralxfr 4261* 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 )
 
TheoremralxfrALT 4262* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. This proof does not use ralxfrd 4257. (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 )
 
Theoremrexxfr 4263* 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 )
 
Theoremrabxfrd 4264* 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 }
 ) )
 
Theoremrabxfr 4265* 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 }
 ) )
 
Theoremreuhypd 4266* 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 )
 
Theoremreuhyp 4267* 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 )
 
Theoremuniexb 4268 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 )
 
Theorempwexb 4269 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 )
 
Theoremuniv 4270 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
 
Theoremeldifpw 4271 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 ) )
 
Theoremop1stb 4272 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
 
Theoremop1stbg 4273 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 )
 
Theoremiunpw 4274* 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)
 
Theoremordon 4275 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
 
Theoremssorduni 4276 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 )
 
Theoremssonuni 4277 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 ) )
 
Theoremssonunii 4278 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 )
 
Theoremonun2 4279 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 )
 
Theoremonun2i 4280 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
 
Theoremordsson 4281 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 )
 
Theoremonss 4282 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  On  ->  A  C_  On )
 
Theoremonuni 4283 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
 |-  ( A  e.  On  ->  U. A  e.  On )
 
Theoremorduni 4284 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
 |-  ( Ord  A  ->  Ord  U. A )
 
Theorembm2.5ii 4285* 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 } )
 
Theoremsucexb 4286 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
 |-  ( A  e.  _V  <->  suc  A  e.  _V )
 
Theoremsucexg 4287 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  V  ->  suc  A  e.  _V )
 
Theoremsucex 4288 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
 |-  A  e.  _V   =>    |-  suc  A  e.  _V
 
Theoremordsucim 4289 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
 |-  ( Ord  A  ->  Ord 
 suc  A )
 
Theoremsuceloni 4290 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 )
 
Theoremordsucg 4291 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
 |-  ( A  e.  _V  ->  ( Ord  A  <->  Ord  suc  A )
 )
 
Theoremsucelon 4292 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
 |-  ( A  e.  On  <->  suc  A  e.  On )
 
Theoremordsucss 4293 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 )
 )
 
Theoremordelsuc 4294 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 )
 )
 
Theoremonsucssi 4295 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 )
 
Theoremonsucmin 4296* 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 }
 )
 
Theoremonsucelsucr 4297 Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4318. However, the converse does hold where  B is a natural number, as seen at nnsucelsuc 6200. (Contributed by Jim Kingdon, 17-Jul-2019.)
 |-  ( B  e.  On  ->  ( suc  A  e.  suc 
 B  ->  A  e.  B ) )
 
Theoremonsucsssucr 4298 The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4315. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
 |-  ( ( A  e.  On  /\  Ord  B )  ->  ( suc  A  C_  suc 
 B  ->  A  C_  B ) )
 
Theoremsucunielr 4299 Successor and union. The converse (where  B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4319. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |-  ( suc  A  e.  B  ->  A  e.  U. B )
 
Theoremunon 4300 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
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