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Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonun2i 4101 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 4102 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 4103 An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.)
 |-  ( A  e.  On  ->  ( A  =  (/)  \/  (/)  e.  A ) )
 
Theoremsnsn0non 4104 The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 4271). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 4396. (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 4105* 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 4107 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4108. A version using class notation is uniex 4109.

The union of a class df-uni 3469 should not be confused with the union of two classes df-un 2835. Their relationship is shown in unipr 3482. (Contributed by NM, 23-Dec-1993.)

 |- 
 E. y A. z
 ( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
 
Theoremzfun 4106* 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 4107* A variant of the Axiom of Union ax-un 4105. 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 4108* 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 4109 The Axiom of Union in class notation. This says that if  A is a set i.e.  A  e.  _V (see isset 2516), 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 4110 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 4111 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 4112 A triple of classes exists. (Contributed by NM, 10-Apr-1994.)
 |- 
 { A ,  B ,  C }  e.  _V
 
Theoremunexb 4113 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 4114 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 4115 A version of unisn 3484 without the  A  e.  _V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)
 |- 
 U. { A }  e.  { (/) ,  A }
 
Theoremunisn3 4116* 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 4117* 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 4118 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 4119 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 4120* 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 ) }
 
Theoremeuuni 4121 If  ph is true for exactly one  x, then  U. {
x  |  ph } is a way to express "the unique element such that  ph is true." Some books use a special symbol such as inverted iota to denote "the unique element such that;" see df-iota 5814. (Contributed by NM, 22-Feb-2004.)
 |-  ( E! x ph  ->  ( ph  <->  U. { x  |  ph
 }  =  x ) )
 
Theoremreuuni1 4122 A way to express "the unique element such that" (restricted quantifier version). (Contributed by NM, 25-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( ( x  e.  A  /\  E! x  e.  A  ph )  ->  ( ph  <->  U. { x  e.  A  |  ph }  =  x ) )
 
Theoremreuuni2f 4123*  U. { x  e.  A  |  ph } is an explicit representation of "the unique element in  A such that  ph." This theorem shows a condition that allows us to represent this element with a class expression  B. The second hypothesis is a weakened bound variable condition that allows hbsbc1g 2696 to be used. (Contributed by NM, 19-Oct-2005.)
 |-  ( y  e.  B  ->  A. x  y  e.  B )   &    |-  ( B  e.  A  ->  ( ps  ->  A. x ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  U. { x  e.  A  |  ph }  =  B ) )
 
Theoremreuuni2 4124*  U. { x  e.  A  |  ph } is an explicit representation of "the unique element in  A such that  ph." (Contributed by NM, 28-Mar-1997.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( ( B  e.  A  /\  E! x  e.  A  ph )  ->  ( ps  <->  U. { x  e.  A  |  ph }  =  B ) )
 
Theoremeuuni2 4125* The unique element such that 
ph. (Contributed by Jeff Madsen, 1-Jun-2011.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  E! x ph )  ->  ( ps 
 <-> 
 U. { x  |  ph
 }  =  A ) )
 
Theoremreuuni3 4126* Derive the property  ch of "the unique element in  A such that  ph " when expressed explicitly as  U. { y  e.  A  |  ps }. (Contributed by NM, 11-Nov-2004.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  U. { y  e.  A  |  ps }  ->  ( ph  <->  ch ) )   =>    |-  ( E! x  e.  A  ph  ->  ch )
 
Theoremreuunisbc 4127* Derive the property of "the unique element in  A such that  ph " when expressed explicitly as  U. { x  e.  A  |  ph }. (Contributed by NM, 11-Nov-2004.)
 |-  ( E! x  e.  A  ph  ->  [ U. { x  e.  A  |  ph
 }  /  x ] ph )
 
Theoremreucl2 4128* Membership law for "the unique element in  A such that  ph." (Contributed by NM, 11-Jun-2005.)
 |-  ( E! x  e.  A  ph  ->  U. { x  e.  A  |  ph
 }  e.  { x  e.  A  |  ph } )
 
Theoremreuuniss 4129* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.)
 |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  U. { x  e.  A  |  ph }  =  U. { x  e.  B  |  ph
 } )
 
Theoremmouniss 4130* Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.)
 |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E* x ( x  e.  B  /\  ph ) )  ->  U. { x  e.  A  |  ph }  =  U. { x  e.  B  |  ph
 } )
 
Theoremreuuniss2 4131* Restriction of a unique element to a smaller class. (Contributed by NM, 20-Oct-2005.)
 |-  ( ( ( A 
 C_  B  /\  A. x  e.  A  ( ph  ->  ps ) )  /\  ( E. x  e.  A  ph 
 /\  E! x  e.  B  ps ) )  ->  U. { x  e.  A  |  ph
 }  =  U. { x  e.  B  |  ps } )
 
Theoremreusn 4132* A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( E! x  e.  A  ph  <->  E. y { x  e.  A  |  ph }  =  { y } )
 
Theoremreusni 4133 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
 |-  B  e.  _V   =>    |-  ( { x  e.  A  |  ph }  =  { B }  ->  E! x  e.  A  ph )
 
Theoremrabsnt 4134* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.)
 |-  B  e.  _V   &    |-  ( x  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( { x  e.  A  |  ph }  =  { B }  ->  ps )
 
Theoremreuunisn 4135 A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
 |-  ( E! x  e.  A  ph  ->  { x  e.  A  |  ph }  =  { U. { x  e.  A  |  ph } }
 )
 
Theoremeusv1 4136* 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 )
 
Theoremeusvhb 4137* Even if  x is free in  A, it is effectively bound when  A ( x ) is single-valued. (Contributed by NM, 14-Oct-2010.)
 |-  ( E! y A. x  y  =  A  ->  ( y  =  A  ->  A. x  y  =  A ) )
 
Theoremeusv2i 4138* 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 E. x  y  =  A )
 
Theoremeusv2 4139* Two ways to express single-valuedness of a class expression  A ( x ). (Contributed by NM, 15-Oct-2010.)
 |-  A  e.  _V   =>    |-  ( E! y E. x  y  =  A 
 <->  E! y A. x  y  =  A )
 
Theoremreusv1lem 4140* Lemma for reusv1 4141. (Contributed by NM, 22-Oct-2010.)
 |-  ( B  =/=  (/)  ->  ( E! x  e.  A  A. y  e.  B  x  =  C  <->  E. x  e.  A  A. y  e.  B  x  =  C ) )
 
Theoremreusv1 4141* Two ways to express single-valuedness of a class expression  C ( y ). (Contributed by NM, 16-Dec-2012.)
 |-  ( 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 4142* Lemma for reusv2 4147. (Contributed by NM, 22-Oct-2010.)
 |-  ( A  =/=  (/)  ->  ( E! x A. y  e.  A  x  =  B  <->  E. x A. y  e.  A  x  =  B ) )
 
Theoremreusv2lem2 4143* Lemma for reusv2 4147. (Contributed by NM, 27-Oct-2010.)
 |-  ( E! x A. y  e.  A  x  =  B  ->  E! x E. y  e.  A  x  =  B )
 
Theoremreusv2lem3 4144* Lemma for reusv2 4147. (Contributed by NM, 14-Dec-2012.)
 |-  ( A. y  e.  A  B  e.  _V  ->  ( E! x E. y  e.  A  x  =  B  <->  E! x A. y  e.  A  x  =  B ) )
 
Theoremreusv2lem4 4145* Lemma for reusv2 4147. (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 4146* Lemma for reusv2 4147. (Contributed by NM, 4-Jan-2013.)
 |-  ( ( 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 4147* 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.)
 |-  ( ( 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 4148* 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 4149* Two ways to express single-valuedness of a class expression  C ( y ). See reusv1 4141 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 4150* 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 4151* 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 4152* 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 3539). (Contributed by NM, 30-Oct-2010.) (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 4153* Two ways to express single-valuedness of a class expression  C ( y ). Note that  U. A  =  |^| A means  A is a singleton (uniintsn 3539). (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 ) )
 
Theoremreusv8OLD 4154* Two ways to express single-valuedness of a class expression  C ( y ). Note that  U. A  =  |^| A means  A is a singleton (uniintsn 3539). (Contributed by NM, 16-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( y  e.  B  /\  ph )  ->  C  e.  A )   =>    |-  ( ( U. A  =/=  |^| 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 ) ) )
 
Theoremeusvobj1 4155* Specify the same object in two ways when class  B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010.)
 |-  B  e.  _V   =>    |-  ( E! x E. y  e.  A  x  =  B  ->  U.
 { x  |  E. y  e.  A  x  =  B }  =  U. { x  |  A. y  e.  A  x  =  B } )
 
Theoremeusvobj2 4156* Specify the same property in two ways when class  B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010.)
 |-  B  e.  _V   =>    |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B  <->  A. y  e.  A  x  =  B )
 )
 
Theoremalxfr 4157* 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 4158* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 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  ->  (
 A. x  e.  B  ps 
 <-> 
 A. y  e.  C  ch ) )
 
Theoremrexxfrd 4159* 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 4160* 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 4161* 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  ->  ( E. x  e.  B  ps 
 <-> 
 E. y  e.  C  ch ) )
 
Theoremralxfr 4162* 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 4163* 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 4164* 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 4165* Class builder membership after substituting an expression  A (containing  y) for  x in the class expression  ch. (Contributed by NM, 16-Jan-2012.)
 |-  ( z  e.  B  ->  A. y  z  e.  B )   &    |-  ( z  e.  C  ->  A. y  z  e.  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 4166* Class builder membership after substituting an expression  A (containing  y) for  x in the class expression  ph. (Contributed by NM, 10-Jun-2005.)
 |-  ( z  e.  B  ->  A. y  z  e.  B )   &    |-  ( z  e.  C  ->  A. y  z  e.  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 4167* 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 4168* 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 4169* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Use reuhypd 4171 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 4170* Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Use reuhyp 4172 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 4171* A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 6541. (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 4172* A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4170. (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 )
 
Theoremreuunixfr 4173* Change the variable  x in the expression for "the unique 
A such that  ph " to another variable  y contained in expression 
B. Use reuhyp 4172 to eliminate the last hypothesis. (Contributed by NM, 13-Jun-2005.)
 |-  ( z  e.  C  ->  A. y  z  e.  C )   &    |-  ( y  e.  A  ->  B  e.  A )   &    |-  ( U. {
 y  e.  A  |  ps }  e.  A  ->  C  e.  A )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  U. {
 y  e.  A  |  ps }  ->  B  =  C )   &    |-  ( x  e.  A  ->  E! y  e.  A  x  =  B )   =>    |-  ( E! x  e.  A  ph  ->  U. { x  e.  A  |  ph
 }  =  C )
 
Theoremuniexb 4174 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 4175 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 4176 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 4177 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 ) )
 
Theoremelpwun 4178 Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.)
 |-  C  e.  _V   =>    |-  ( A  e.  ~P ( B  u.  C ) 
 <->  ( A  \  C )  e.  ~P B )
 
Theoremelpwunsn 4179 Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.)
 |-  ( A  e.  ( ~P ( B  u.  { C } )  \  ~P B )  ->  C  e.  A )
 
Theoremop1stb 4180 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 4780 to extract the second member, op1sta 4778 for an alternate version, and op1st 5664 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 |^| |^| <. A ,  B >.  =  A
 
Theoremiunpw 4181* 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 )
 
Theoremfr3nr 4182 A well-founded relation has no 3-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 10-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
 |-  ( ( R  Fr  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  -.  ( B R C  /\  C R D  /\  D R B ) )
 
Theoremepne3 4183 A set well-founded by epsilon contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
 |-  ( (  _E  Fr  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  -.  ( B  e.  C  /\  C  e.  D  /\  D  e.  B )
 )
 
Theoremdfwe2 4184* Alternate definition of well-ordering. Definition 6.24(2) of [TakeutiZaring] p. 30. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( R  We  A  <->  ( R  Fr  A  /\  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) ) )
 
2.4.2  Ordinals (continued)
 
Theoremordon 4185 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
 
Theoremepweon 4186 The epsilon relation well-orders the class of ordinal numbers. Proposition 4.8(g) of [Mendelson] p. 244. (Contributed by NM, 1-Nov-2003.)
 |- 
 _E  We  On
 
Theoremonprc 4187 No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4185), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.)
 |- 
 -.  On  e.  _V
 
Theoremssorduni 4188 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 4189 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 4190 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 )
 
Theoremordeleqon 4191 A way to express the ordinal property of a class in terms of the class of ordinal numbers. Corollary 7.14 of [TakeutiZaring] p. 38 and its converse. (Contributed by NM, 1-Jun-2003.)
 |-  ( Ord  A  <->  ( A  e.  On  \/  A  =  On ) )
 
Theoremordsson 4192 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.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( Ord  A  ->  A 
 C_  On )
 
Theoremonss 4193 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  On  ->  A  C_  On )
 
Theoremssonprc 4194 Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.)
 |-  ( A  C_  On  ->  ( A  e/  _V  <->  U. A  =  On )
 )
 
Theoremonuni 4195 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
 |-  ( A  e.  On  ->  U. A  e.  On )
 
Theoremorduni 4196 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
 |-  ( Ord  A  ->  Ord  U. A )
 
Theoremonint 4197 The intersection (infimum) of a non-empty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45. (Contributed by NM, 31-Jan-1997.)
 |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
 
Theoremonint0 4198 The intersection of a class of ordinal numbers is zero iff the class contains zero. (Contributed by NM, 24-Apr-2004.)
 |-  ( A  C_  On  ->  ( |^| A  =  (/)  <->  (/)  e.  A ) )
 
Theoremonssmin 4199* A non-empty class of ordinal numbers has a smallest member. Exercise 9 of [TakeutiZaring] p. 40. (Contributed by NM, 3-Oct-2003.)
 |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  x  C_  y )
 
Theoremonminsb 4200 If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  = 
 |^| { x  e.  On  |  ph }  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x  e.  On  ph  ->  ps )
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