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Theorem List for Metamath Proof Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelelsuc 4401 Membership in a successor. (Contributed by NM, 20-Jun-1998.)
 |-  ( A  e.  B  ->  A  e.  suc  B )
 
Theoremsucel 4402* 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 4403 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
 |- 
 suc  (/)  =  { (/) }
 
Theoremsucprc 4404 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
 |-  ( -.  A  e.  _V 
 ->  suc  A  =  A )
 
Theoremunisuc 4405 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 )
 
Theoremsssucid 4406 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 4407 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 4408 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
 
Theoremnsuceq0 4409 No successor is empty. (Contributed by NM, 3-Apr-1995.)
 |- 
 suc  A  =/=  (/)
 
Theoremeqelsuc 4410 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 4411* 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 4412 The successor of a transtive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
 |-  ( Tr  A  ->  Tr 
 suc  A )
 
Theoremtrsuc 4413 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 )
 
Theoremtrsuc2OLD 4414 Obsolete proof of suctr 4412 as of 5-Apr-2016. The successor of a transitive set is transitive. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr 
 suc  A )
 
Theoremtrsucss 4415 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 )
 )
 
Theoremordsssuc 4416 A subset of an ordinal belongs to its successor. (Contributed by NM, 28-Nov-2003.)
 |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  A  e.  suc  B )
 )
 
Theoremonsssuc 4417 A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B 
 <->  A  e.  suc  B ) )
 
Theoremordsssuc2 4418 An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( ( Ord  A  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B )
 )
 
Theoremonmindif 4419 When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.)
 |-  ( ( A  C_  On  /\  B  e.  On )  ->  B  e.  |^| ( A  \  suc  B ) )
 
Theoremordnbtwn 4420 There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 21-Jun-1998.)
 |-  ( Ord  A  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
 
Theoremonnbtwn 4421 There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 9-Jun-1994.)
 |-  ( A  e.  On  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
 
Theoremsucssel 4422 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 ) )
 
Theoremorddif 4423 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
 |-  ( Ord  A  ->  A  =  ( suc  A  \  { A } )
 )
 
Theoremorduniss 4424 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
 |-  ( Ord  A  ->  U. A  C_  A )
 
Theoremordtri2or 4425 A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  B  C_  A )
 )
 
Theoremordtri2or2 4426 A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  \/  B  C_  A ) )
 
Theoremordtri2or3 4427 A consequence of total ordering for ordinal classes. Similar to ordtri2or2 4426. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  ( A  i^i  B )  \/  B  =  ( A  i^i  B ) ) )
 
Theoremordelinel 4428 The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.)
 |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( A  i^i  B )  e.  C  <->  ( A  e.  C  \/  B  e.  C ) ) )
 
Theoremordssun 4429 Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)
 |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  C_  ( B  u.  C )  <->  ( A  C_  B  \/  A  C_  C ) ) )
 
Theoremordequn 4430 The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
 |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  =  ( B  u.  C )  ->  ( A  =  B  \/  A  =  C ) ) )
 
Theoremordun 4431 The maximum (i.e. union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
 |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  u.  B ) )
 
Theoremordunisssuc 4432 A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.)
 |-  ( ( A  C_  On  /\  Ord  B )  ->  ( U. A  C_  B 
 <->  A  C_  suc  B ) )
 
Theoremsuc11 4433 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 ) )
 
Theoremonordi 4434 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  On   =>    |-  Ord  A
 
Theoremontrci 4435 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  On   =>    |-  Tr  A
 
Theoremonirri 4436 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
 
Theoremoneli 4437 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 4438 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 )
 
Theoremonssneli 4439 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
 |-  A  e.  On   =>    |-  ( A  C_  B  ->  -.  B  e.  A )
 
Theoremonssnel2i 4440 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
 |-  A  e.  On   =>    |-  ( B  C_  A  ->  -.  A  e.  B )
 
Theoremonelini 4441 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 4442 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 4443 An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
 |-  A  e.  On   =>    |-  U. suc  A  =  A
 
Theoremonsseli 4444 Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.)
 |-  A  e.  On   &    |-  B  e.  On   =>    |-  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
 )
 
Theoremonun2i 4445 The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)
 |-  A  e.  On   &    |-  B  e.  On   =>    |-  ( A  u.  B )  e.  On
 
Theoremunizlim 4446 An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
 |-  ( Ord  A  ->  ( A  =  U. A  <->  ( A  =  (/)  \/  Lim  A ) ) )
 
Theoremon0eqel 4447 An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.)
 |-  ( A  e.  On  ->  ( A  =  (/)  \/  (/)  e.  A ) )
 
Theoremsnsn0non 4448 The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 4597). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 4722. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |- 
 -.  { { (/) } }  e.  On
 
2.4  ZF Set Theory - add the Axiom of Union
 
2.4.1  Introduce the Axiom of Union
 
Axiomax-un 4449* Axiom of Union. An axiom of 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 4451 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4452. A version using class notation is uniex 4453.

The union of a class df-uni 3769 should not be confused with the union of two classes df-un 3099. Their relationship is shown in unipr 3782. (Contributed by NM, 23-Dec-1993.)

 |- 
 E. y A. z
 ( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
 
Theoremzfun 4450* Axiom of Union expressed with 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 4451* A variant of the Axiom of Union ax-un 4449. 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 4452* 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 4453 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
 
Theoremuniexg 4454 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 4455 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
 
Theoremtpex 4456 A triple of classes exists. (Contributed by NM, 10-Apr-1994.)
 |- 
 { A ,  B ,  C }  e.  _V
 
Theoremunexb 4457 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 4458 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 )
 
Theoremunisn2 4459 A version of unisn 3784 without the  A  e.  _V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)
 |- 
 U. { A }  e.  { (/) ,  A }
 
Theoremunisn3 4460* 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 4461* 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
 
Theoremdifex2 4462 If the subtrahend of a class difference exists, then the minuend exists iff the difference exists. (Contributed by NM, 12-Nov-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( B  e.  C  ->  ( A  e.  _V  <->  ( A  \  B )  e. 
 _V ) )
 
Theoremopeluu 4463 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 4464* 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 4465* 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 4466* 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 4467* 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 4468* 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 4469* 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 4470* 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 4471* 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 ) ) )
 
Theoremreusv2lem1 4472* Lemma for reusv2 4477. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
 |-  ( A  =/=  (/)  ->  ( E! x A. y  e.  A  x  =  B  <->  E. x A. y  e.  A  x  =  B ) )
 
Theoremreusv2lem2 4473* Lemma for reusv2 4477. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
 |-  ( E! x A. y  e.  A  x  =  B  ->  E! x E. y  e.  A  x  =  B )
 
Theoremreusv2lem3 4474* Lemma for reusv2 4477. (Contributed by NM, 14-Dec-2012.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
 |-  ( A. y  e.  A  B  e.  _V  ->  ( E! x E. y  e.  A  x  =  B  <->  E! x A. y  e.  A  x  =  B ) )
 
Theoremreusv2lem4 4475* Lemma for reusv2 4477. (Contributed by NM, 13-Dec-2012.)
 |-  ( E! x  e.  A  E. y  e.  B  ( ph  /\  x  =  C )  <->  E! x A. y  e.  B  ( ( C  e.  A  /\  ph )  ->  x  =  C ) )
 
Theoremreusv2lem5 4476* Lemma for reusv2 4477. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
 |-  ( ( A. y  e.  B  C  e.  A  /\  B  =/=  (/) )  ->  ( E! x  e.  A  E. y  e.  B  x  =  C  <->  E! x  e.  A  A. y  e.  B  x  =  C ) )
 
Theoremreusv2 4477* Two ways to express single-valuedness of a class expression  C
( y ) that is constant for those  y  e.  B such that  ph. The first antecedent ensures that the constant value belongs to the existential uniqueness domain  A, and the second ensures that  C ( y ) is evaluated for at least one  y. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
 |-  ( ( A. y  e.  B  ( ph  ->  C  e.  A )  /\  E. y  e.  B  ph )  ->  ( E! x  e.  A  E. y  e.  B  ( ph  /\  x  =  C )  <->  E! x  e.  A  A. y  e.  B  (
 ph  ->  x  =  C ) ) )
 
Theoremreusv3i 4478* 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 4479* Two ways to express single-valuedness of a class expression  C ( y ). See reusv1 4471 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 ) ) )
 
Theoremeusv4 4480* Two ways to express single-valuedness of a class expression  B ( x ). (Contributed by NM, 27-Oct-2010.)
 |-  B  e.  _V   =>    |-  ( E! x E. y  e.  A  x  =  B  <->  E! x A. y  e.  A  x  =  B )
 
Theoremreusv5OLD 4481* Two ways to express single-valuedness of a class expression  C ( y ). (Contributed by NM, 16-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( B  =/=  (/)  ->  ( E! x  e.  A  A. y  e.  B  x  =  C  <->  E. x  e.  A  A. y  e.  B  x  =  C ) )
 
Theoremreusv6OLD 4482* Two ways to express single-valuedness of a class expression  C ( y ). The converse does not hold. Note that  U. A  =  |^| A means  A is a singleton (uniintsn 3840). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( U. A  =/=  |^| A  \/  B  =/= 
 (/) )  ->  ( E! x  e.  A  A. y  e.  B  x  =  C  ->  E! x  e.  A  E. y  e.  B  x  =  C ) )
 
Theoremreusv7OLD 4483* Two ways to express single-valuedness of a class expression  C ( y ). Note that  U. A  =  |^| A means  A is a singleton (uniintsn 3840). (Contributed by NM, 14-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( y  e.  B  ->  C  e.  A )   =>    |-  ( ( U. A  =/=  |^| A  \/  B  =/= 
 (/) )  ->  ( E! x  e.  A  E. y  e.  B  x  =  C  <->  E! x  e.  A  A. y  e.  B  x  =  C ) )
 
Theoremalxfr 4484* 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 4485* 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 4486* 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 4487* 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 4488* 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 4489* 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 4490* 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.) (Proof modification 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 4491* 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 4492* 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 4493* 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 }
 ) )
 
Theoremreuxfr2d 4494* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 16-Jan-2012.)
 |-  ( ( ph  /\  y  e.  B )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E* y ( y  e.  B  /\  x  =  A ) )   =>    |-  ( ph  ->  ( E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ps )  <->  E! y  e.  B  ps ) )
 
Theoremreuxfr2 4495* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 14-Nov-2004.)
 |-  ( y  e.  B  ->  A  e.  B )   &    |-  ( x  e.  B  ->  E* y ( y  e.  B  /\  x  =  A ) )   =>    |-  ( E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ph )  <->  E! y  e.  B  ph )
 
Theoremreuxfrd 4496* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Use reuhypd 4498 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.)
 |-  ( ( ph  /\  y  e.  B )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E! y  e.  B  x  =  A )   &    |-  ( x  =  A  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E! x  e.  B  ps 
 <->  E! y  e.  B  ch ) )
 
Theoremreuxfr 4497* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Use reuhyp 4499 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.)
 |-  ( y  e.  B  ->  A  e.  B )   &    |-  ( x  e.  B  ->  E! y  e.  B  x  =  A )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E! x  e.  B  ph  <->  E! y  e.  B  ps )
 
Theoremreuhypd 4498* A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 6269. (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 4499* A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4497. (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 4500 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 )
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