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Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremdtruALT 4101* A version of dtru 4095 ("two things exist") with a shorter proof that uses more axioms but may be easier to understand.

Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that  x and  y be distinct. Specifically, theorem cla4ev 2812 requires that  x must not occur in the subexpression  -.  y  =  { (/) } in step 4 nor in the subexpression  -.  y  =  (/) in step 9. The proof verifier will require that  x and  y be in a distinct variable group to ensure this. You can check this by deleting the $d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation. (Contributed by NM, 15-Jul-1994.) (Proof modification is discouraged.)

 |- 
 -.  A. x  x  =  y
 
Theoremdtrucor 4102* Corollary of dtru 4095. This example illustrates the danger of blindly trusting the standard Deduction Theorem without accounting for free variables: the theorem form of this deduction is not valid, as shown by dtrucor2 4103. (Contributed by NM, 27-Jun-2002.)
 |-  x  =  y   =>    |-  x  =/=  y
 
Theoremdtrucor2 4103 The theorem form of the deduction dtrucor 4102 leads to a contradiction, as mentioned in the "Wrong!" example at http://us.metamath.org/mpegif/mmdeduction.html#bad. (Contributed by NM, 20-Oct-2007.)
 |-  ( x  =  y 
 ->  x  =/=  y
 )   =>    |-  ( ph  /\  -.  ph )
 
Theoremdvdemo1 4104* Demonstration of a theorem (scheme) that requires (meta)variables  x and  y to be distinct, but no others. It bundles the theorem schemes  E. x ( x  =  y  ->  x  e.  x ) and  E. x ( x  =  y  ->  y  e.  x ). Compare dvdemo2 4105. ("Bundles" is a term introduced by Raph Levien.) (Contributed by NM, 1-Dec-2006.)
 |- 
 E. x ( x  =  y  ->  z  e.  x )
 
Theoremdvdemo2 4105* Demonstration of a theorem (scheme) that requires (meta)variables  x and  z to be distinct, but no others. It bundles the theorem schemes  E. x ( x  =  x  -> 
z  e.  x ) and 
E. x ( x  =  y  ->  y  e.  x ). Compare dvdemo1 4104. (Contributed by NM, 1-Dec-2006.)
 |- 
 E. x ( x  =  y  ->  z  e.  x )
 
2.3.2  Derive the Axiom of Pairing
 
Theoremzfpair 4106 The Axiom of Pairing of Zermelo-Fraenkel set theory. Axiom 2 of [TakeutiZaring] p. 15. In some textbooks this is stated as a separate axiom; here we show it is redundant since it can be derived from the other axioms.

This theorem should not be referenced by any proof other than axpr 4107. Instead, use zfpair2 4109 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)

 |- 
 { x ,  y }  e.  _V
 
Theoremaxpr 4107* Unabbreviated version of the Axiom of Pairing of ZF set theory, derived as a theorem from the other axioms.

This theorem should not be referenced by any proof. Instead, use ax-pr 4108 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)

 |- 
 E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
 
Axiomax-pr 4108* The Axiom of Pairing of ZF set theory. It was derived as theorem axpr 4107 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 14-Nov-2006.)
 |- 
 E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
 
Theoremzfpair2 4109 Derive the abbreviated version of the Axiom of Pairing from ax-pr 4108. See zfpair 4106 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)
 |- 
 { x ,  y }  e.  _V
 
Theoremsnex 4110 A singleton is a set. Theorem 7.13 of [Quine] p. 51, proved using Extensionality, Separation, and Pairing. See also snexALT 4090. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) (Proof modification is discouraged.)
 |- 
 { A }  e.  _V
 
Theoremprex 4111 The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51. By virtue of its definition, an unordered pair remains a set (even though no longer a pair) even when its components are proper classes (see prprc 3642), so we can dispense with hypotheses requiring them to be sets. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A ,  B }  e.  _V
 
TheoremelALT 4112* Every set is an element of some other set. This has a shorter proof than el 4086 but uses more axioms. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.)
 |- 
 E. y  x  e.  y
 
TheoremdtruALT2 4113* An alternative proof of dtru 4095 ("two things exist") using ax-pr 4108 instead of ax-pow 4082. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.)
 |- 
 -.  A. x  x  =  y
 
Theoremsnelpwi 4114 A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
 |-  ( A  e.  B  ->  { A }  e.  ~P B )
 
Theoremsnelpw 4115 A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
 |-  A  e.  _V   =>    |-  ( A  e.  B 
 <->  { A }  e.  ~P B )
 
Theoremrext 4116* A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
 |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  x  =  y )
 
Theoremsspwb 4117 Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
 |-  ( A  C_  B  <->  ~P A  C_  ~P B )
 
Theoremunipw 4118 A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)
 |- 
 U. ~P A  =  A
 
Theorempwel 4119 Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
 |-  ( A  e.  B  ->  ~P A  e.  ~P ~P U. B )
 
Theorempwtr 4120 A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
 |-  ( Tr  A  <->  Tr  ~P A )
 
Theoremssextss 4121* An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
 |-  ( A  C_  B  <->  A. x ( x  C_  A  ->  x  C_  B ) )
 
Theoremssext 4122* An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
 |-  ( A  =  B  <->  A. x ( x  C_  A 
 <->  x  C_  B )
 )
 
Theoremnssss 4123* Negation of subclass relationship. Compare nss 3157. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( -.  A  C_  B 
 <-> 
 E. x ( x 
 C_  A  /\  -.  x  C_  B ) )
 
Theorempweqb 4124 Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
 |-  ( A  =  B  <->  ~P A  =  ~P B )
 
Theoremintid 4125* The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
 |-  A  e.  _V   =>    |-  |^| { x  |  A  e.  x }  =  { A }
 
Theoremmoabex 4126 "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)
 |-  ( E* x ph  ->  { x  |  ph }  e.  _V )
 
Theoremeuabex 4127 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
 |-  ( E! x ph  ->  { x  |  ph }  e.  _V )
 
Theoremnnullss 4128* A non-empty class (even if proper) has a non-empty subset. (Contributed by NM, 23-Aug-2003.)
 |-  ( A  =/=  (/)  ->  E. x ( x  C_  A  /\  x  =/=  (/) ) )
 
Theoremexss 4129* Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
 |-  ( E. x  e.  A  ph  ->  E. y
 ( y  C_  A  /\  E. x  e.  y  ph ) )
 
Theoremopex 4130 An ordered pair of classes is a set. Exercise 7 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |- 
 <. A ,  B >.  e. 
 _V
 
Theoremotex 4131 An ordered triple of classes is a set. (Contributed by NM, 3-Apr-2015.)
 |- 
 <. A ,  B ,  C >.  e.  _V
 
Theoremelop 4132 An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  e.  <. B ,  C >. 
 <->  ( A  =  { B }  \/  A  =  { B ,  C } ) )
 
Theoremopi1 4133 One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 { A }  e.  <. A ,  B >.
 
Theoremopi2 4134 One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 { A ,  B }  e.  <. A ,  B >.
 
2.3.3  Ordered pair theorem
 
Theoremopnz 4135 An ordered pair is nonempty iff the arguments are sets. (Contributed by NM, 24-Jan-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( <. A ,  B >.  =/=  (/)  <->  ( A  e.  _V 
 /\  B  e.  _V ) )
 
Theoremopnzi 4136 An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 <. A ,  B >.  =/=  (/)
 
Theoremopth1 4137 Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  =  <. C ,  D >.  ->  A  =  C )
 
Theoremopth 4138 The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that  C and  D are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
 ( A  =  C  /\  B  =  D ) )
 
Theoremopthg 4139 Ordered pair theorem.  C and  D are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >. 
 <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremopthg2 4140 Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( C  e.  V  /\  D  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >. 
 <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremopth2 4141 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
 |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
 ( A  =  C  /\  B  =  D ) )
 
Theoremotth2 4142 Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  R  e.  _V   =>    |-  ( <.
 <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )
 
Theoremotth 4143 Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  R  e.  _V   =>    |-  ( <. A ,  B ,  R >.  =  <. C ,  D ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S )
 )
 
Theoremeqvinop 4144* A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  =  <. B ,  C >.  <->  E. x E. y
 ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. ) )
 
Theoremcopsexg 4145* Substitution of class  A for ordered pair  <. x ,  y
>.. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  ( A  =  <. x ,  y >.  ->  ( ph 
 <-> 
 E. x E. y
 ( A  =  <. x ,  y >.  /\  ph )
 ) )
 
Theoremcopsex2t 4146* Closed theorem form of copsex2g 4147. (Contributed by NM, 17-Feb-2013.)
 |-  ( ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )  /\  ( A  e.  V  /\  B  e.  W ) )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph )  <->  ps ) )
 
Theoremcopsex2g 4147* Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x E. y (
 <. A ,  B >.  = 
 <. x ,  y >.  /\  ph )  <->  ps ) )
 
Theoremcopsex4g 4148* An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
 |-  ( ( ( x  =  A  /\  y  =  B )  /\  (
 z  =  C  /\  w  =  D )
 )  ->  ( ph  <->  ps ) )   =>    |-  ( ( ( A  e.  R  /\  B  e.  S )  /\  ( C  e.  R  /\  D  e.  S )
 )  ->  ( E. x E. y E. z E. w ( ( <. A ,  B >.  =  <. x ,  y >.  /\  <. C ,  D >.  =  <. z ,  w >. )  /\  ph )  <->  ps ) )
 
Theorem0nelop 4149 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 -.  (/)  e.  <. A ,  B >.
 
Theoremopeqex 4150 Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
 |-  ( <. A ,  B >.  =  <. C ,  D >.  ->  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( C  e.  _V 
 /\  D  e.  _V ) ) )
 
Theoremoteqex2 4151 Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.)
 |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( C  e.  _V  <->  T  e.  _V ) )
 
Theoremoteqex 4152 Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  <->  ( R  e.  _V 
 /\  S  e.  _V  /\  T  e.  _V )
 ) )
 
Theoremopcom 4153 An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  =  <. B ,  A >.  <->  A  =  B )
 
Theoremmoop2 4154* "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  B  e.  _V   =>    |-  E* x  A  =  <. B ,  x >.
 
Theoremopeqsn 4155 Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.  =  { C }  <->  ( A  =  B  /\  C  =  { A } ) )
 
Theoremopeqpr 4156 Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( <. A ,  B >.  =  { C ,  D }  <->  ( ( C  =  { A }  /\  D  =  { A ,  B } )  \/  ( C  =  { A ,  B }  /\  D  =  { A } ) ) )
 
Theoremmosubopt 4157* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
 |-  ( A. y A. z E* x ph  ->  E* x E. y E. z ( A  =  <. y ,  z >.  /\  ph ) )
 
Theoremmosubop 4158* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
 |- 
 E* x ph   =>    |- 
 E* x E. y E. z ( A  =  <. y ,  z >.  /\  ph )
 
Theoremeuop2 4159* Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004.)
 |-  A  e.  _V   =>    |-  ( E! x E. y ( x  = 
 <. A ,  y >.  /\  ph )  <->  E! y ph )
 
Theoremeuotd 4160* Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  _V )   &    |-  ( ph  ->  ( ps 
 <->  ( a  =  A  /\  b  =  B  /\  c  =  C ) ) )   =>    |-  ( ph  ->  E! x E. a E. b E. c ( x  =  <. a ,  b ,  c >.  /\  ps )
 )
 
Theoremopthwiener 4161 Justification theorem for the ordered pair definition in Norbert Wiener, "A simplification of the logic of relations," Proc. of the Cambridge Philos. Soc., 1914, vol. 17, pp.387-390. It is also shown as a definition in [Enderton] p. 36 and as Exercise 4.8(b) of [Mendelson] p. 230. It is meaningful only for classes that exist as sets (i.e. are not proper classes). See df-op 3553 for other ordered pair definitions. (Contributed by NM, 28-Sep-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( { { { A } ,  (/) } ,  { { B } } }  =  { { { C } ,  (/) } ,  { { D } } } 
 <->  ( A  =  C  /\  B  =  D ) )
 
Theoremuniop 4162 The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 U. <. A ,  B >.  =  { A ,  B }
 
Theoremuniopel 4163 Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  C  ->  U. <. A ,  B >.  e.  U. C )
 
2.3.4  Ordered-pair class abstractions (cont.)
 
Theoremopabid 4164 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( <. x ,  y >.  e.  { <. x ,  y >.  |  ph }  <->  ph )
 
Theoremelopab 4165* Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
 |-  ( A  e.  { <. x ,  y >.  | 
 ph }  <->  E. x E. y
 ( A  =  <. x ,  y >.  /\  ph )
 )
 
TheoremopelopabsbOLD 4166* The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.)
 |-  ( <. z ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ w  /  y ] [ z  /  x ] ph )
 
TheorembrabsbOLD 4167* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.) (New usage is discouraged.)
 |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( z R w  <->  [ w  /  y ] [ z  /  x ] ph )
 
Theoremopelopabsb 4168* The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
 |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ].
 [. B  /  y ]. ph )
 
Theorembrabsb 4169* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
 |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( A R B  <->  [. A  /  x ].
 [. B  /  y ]. ph )
 
Theoremopelopabt 4170* Closed theorem form of opelopab 4179. (Contributed by NM, 19-Feb-2013.)
 |-  ( ( A. x A. y ( x  =  A  ->  ( ph  <->  ps ) )  /\  A. x A. y ( y  =  B  ->  ( ps  <->  ch ) )  /\  ( A  e.  V  /\  B  e.  W ) )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch ) )
 
Theoremopelopabga 4171* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ph }  <->  ps ) )
 
Theorembrabga 4172* The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
 
Theoremopelopab2a 4173* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D )  /\  ph ) } 
 <->  ps ) )
 
Theoremopelopaba 4174* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ps )
 
Theorembraba 4175* The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( A R B 
 <->  ps )
 
Theoremopelopabg 4176* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ph }  <->  ch ) )
 
Theorembrabg 4177* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  R  =  { <. x ,  y >.  |  ph }   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
 
Theoremopelopab2 4178* Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e. 
 { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D )  /\  ph ) } 
 <->  ch ) )
 
Theoremopelopab 4179* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
 
Theorembrab 4180* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   &    |-  R  =  { <. x ,  y >.  | 
 ph }   =>    |-  ( A R B  <->  ch )
 
Theoremopelopabaf 4181* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4179 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |- 
 F/ x ps   &    |-  F/ y ps   &    |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ps )
 
Theoremopelopabf 4182* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4179 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by NM, 19-Dec-2008.)
 |- 
 F/ x ps   &    |-  F/ y ch   &    |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch )
 
Theoremssopab2 4183 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)
 |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  {
 <. x ,  y >.  |  ps } )
 
Theoremssopab2b 4184 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  ( { <. x ,  y >.  |  ph }  C_  {
 <. x ,  y >.  |  ps }  <->  A. x A. y
 ( ph  ->  ps )
 )
 
Theoremssopab2i 4185 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
 |-  ( ph  ->  ps )   =>    |-  { <. x ,  y >.  |  ph } 
 C_  { <. x ,  y >.  |  ps }
 
Theoremssopab2dv 4186* Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  {
 <. x ,  y >.  |  ps }  C_  { <. x ,  y >.  |  ch } )
 
Theoremeqopab2b 4187 Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  ps }  <->  A. x A. y
 ( ph  <->  ps ) )
 
Theoremopabn0 4188 Non-empty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.)
 |-  ( { <. x ,  y >.  |  ph }  =/=  (/)  <->  E. x E. y ph )
 
Theoremiunopab 4189* Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  U_ z  e.  A  { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  E. z  e.  A  ph }
 
2.3.5  Power class of union and intersection
 
Theorempwin 4190 The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
 |- 
 ~P ( A  i^i  B )  =  ( ~P A  i^i  ~P B )
 
Theorempwunss 4191 The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
 |-  ( ~P A  u.  ~P B )  C_  ~P ( A  u.  B )
 
Theorempwssun 4192 The power class of the union of two classes is a subset of the union of their power classes, iff one class is a subclass of the other. Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
 |-  ( ( A  C_  B  \/  B  C_  A ) 
 <->  ~P ( A  u.  B )  C_  ( ~P A  u.  ~P B ) )
 
Theorempwundif 4193 Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Tirix, 20-Dec-2016.)
 |- 
 ~P ( A  u.  B )  =  (
 ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )
 
TheorempwundifOLD 4194 Break up the power class of a union into a union of smaller classes. Obsolete as of 20-Dec-2016. (Contributed by NM, 25-Mar-2007.) (New usage is discouraged.)
 |- 
 ~P ( A  u.  B )  =  (
 ( ~P ( A  u.  B )  \  ~P A )  u.  ~P A )
 
Theorempwun 4195 The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by NM, 23-Nov-2003.)
 |-  ( ( A  C_  B  \/  B  C_  A ) 
 <->  ~P ( A  u.  B )  =  ( ~P A  u.  ~P B ) )
 
2.3.6  Epsilon and identity relations
 
Syntaxcep 4196 Extend class notation to include the epsilon relation.
 class  _E
 
Syntaxcid 4197 Extend the definition of a class to include identity relation.
 class  _I
 
Definitiondf-eprel 4198* Define the epsilon relation. Similar to Definition 6.22 of [TakeutiZaring] p. 30. The epsilon relation and set membership are the same, that is,  ( A  _E  B  <->  A  e.  B ) when  B is a set by epelg 4199. Thus,  5  _E  { 1 ,  5 } (ex-eprel 20633). (Contributed by NM, 13-Aug-1995.)
 |- 
 _E  =  { <. x ,  y >.  |  x  e.  y }
 
Theoremepelg 4199 The epsilon relation and membership are the same. General version of epel 4201. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
 
Theoremepelc 4200 The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  B  e.  _V   =>    |-  ( A  _E  B 
 <->  A  e.  B )
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