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Theorem List for Intuitionistic Logic Explorer - 4301-4400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsuc0 4301 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
 |- 
 suc  (/)  =  { (/) }
 
Theoremsucprc 4302 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
 |-  ( -.  A  e.  _V 
 ->  suc  A  =  A )
 
Theoremunisuc 4303 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 4304 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 4305 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 4306 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 4307 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 4308 No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
 |-  ( A  e.  V  ->  suc  A  =/=  (/) )
 
Theoremeqelsuc 4309 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 4310* 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 4311 The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
 |-  ( Tr  A  ->  Tr 
 suc  A )
 
Theoremtrsuc 4312 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 4313 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 4314 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 4315 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
 |-  ( Ord  A  ->  U. A  C_  A )
 
Theoremonordi 4316 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  On   =>    |-  Ord  A
 
Theoremontrci 4317 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  On   =>    |-  Tr  A
 
Theoremoneli 4318 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 4319 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 4320 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 4321 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 4322 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 4323* 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 4325 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4326. A version using class notation is uniex 4327.

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 4017), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 264).

The union of a class df-uni 3705 should not be confused with the union of two classes df-un 3043. Their relationship is shown in unipr 3718. (Contributed by NM, 23-Dec-1993.)

 |- 
 E. y A. z
 ( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
 
Theoremzfun 4324* 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 4325* A variant of the Axiom of Union ax-un 4323. 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 4326* 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 4327 The Axiom of Union in class notation. This says that if  A is a set i.e.  A  e.  _V (see isset 2664), 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
 
Theoremvuniex 4328 The union of a setvar is a set. (Contributed by BJ, 3-May-2021.)
 |- 
 U. x  e.  _V
 
Theoremuniexg 4329 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 4330 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 4331 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 4332 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 4333 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 4334* 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 )
 
Theoremabnexg 4335* 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 5982. 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 4337 and pwnex 4338 respectively, proved from abnex 4336, which is a consequence of abnexg 4335 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 ) )
 
Theoremabnex 4336* Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 4337 and pwnex 4338. See the comment of abnexg 4335. (Contributed by BJ, 2-May-2021.)
 |-  ( A. x ( F  e.  V  /\  x  e.  F )  ->  -.  { y  | 
 E. x  y  =  F }  e.  _V )
 
Theoremsnnex 4337* 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
 
Theorempwnex 4338* The class of all power sets is a proper class. See also snnex 4337. (Contributed by BJ, 2-May-2021.)
 |- 
 { x  |  E. y  x  =  ~P y }  e/  _V
 
Theoremopeluu 4339 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 4340* 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 4341* 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 4342* 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 4343* 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 4344* 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 4345* 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 4346* 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 4347* 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 4348* 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 4349* Two ways to express single-valuedness of a class expression  C ( y ). See reusv1 4347 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 4350* 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 4351* 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 4352* 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 4353* 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 4354* 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 4355* 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 4356* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. This proof does not use ralxfrd 4351. (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 4357* 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 4358* 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 4359* 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 4360* 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 4361* 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 4362 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 4363 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 )
 
Theoremelpwpwel 4364 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 )
 
Theoremuniv 4365 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 4366 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 4367 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 4368 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 4369* 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 4370 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 4371 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 4372 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 4373 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 4374 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 4375 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 4376 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 4377 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 4378 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
 |-  ( A  e.  On  ->  U. A  e.  On )
 
Theoremorduni 4379 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
 |-  ( Ord  A  ->  Ord  U. A )
 
Theorembm2.5ii 4380* 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 4381 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
 |-  ( A  e.  _V  <->  suc  A  e.  _V )
 
Theoremsucexg 4382 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  V  ->  suc  A  e.  _V )
 
Theoremsucex 4383 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
 |-  A  e.  _V   =>    |-  suc  A  e.  _V
 
Theoremordsucim 4384 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
 |-  ( Ord  A  ->  Ord 
 suc  A )
 
Theoremsuceloni 4385 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 4386 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
 |-  ( A  e.  _V  ->  ( Ord  A  <->  Ord  suc  A )
 )
 
Theoremsucelon 4387 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
 |-  ( A  e.  On  <->  suc  A  e.  On )
 
Theoremordsucss 4388 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 4389 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 4390 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 4391* 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 4392 Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4413. However, the converse does hold where  B is a natural number, as seen at nnsucelsuc 6353. (Contributed by Jim Kingdon, 17-Jul-2019.)
 |-  ( B  e.  On  ->  ( suc  A  e.  suc 
 B  ->  A  e.  B ) )
 
Theoremonsucsssucr 4393 The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4410. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
 |-  ( ( A  e.  On  /\  Ord  B )  ->  ( suc  A  C_  suc 
 B  ->  A  C_  B ) )
 
Theoremsucunielr 4394 Successor and union. The converse (where  B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4414. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |-  ( suc  A  e.  B  ->  A  e.  U. B )
 
Theoremunon 4395 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
 
Theoremonuniss2 4396* 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 )
 
Theoremlimon 4397 The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
 |- 
 Lim  On
 
Theoremordunisuc2r 4398* 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 ) )
 
Theoremonssi 4399 An ordinal number is a subset of 
On. (Contributed by NM, 11-Aug-1994.)
 |-  A  e.  On   =>    |-  A  C_  On
 
Theoremonsuci 4400 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
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