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Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssext 4101* 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 4102* Negation of subclass relationship. Compare nss 3137. (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 4103 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 4104* 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 4105 "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)
 |-  ( E* x ph  ->  { x  |  ph }  e.  _V )
 
Theoremeuabex 4106 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
 |-  ( E! x ph  ->  { x  |  ph }  e.  _V )
 
Theoremnnullss 4107* 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 4108* 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 4109 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 4110 An ordered triple of classes is a set. (Contributed by NM, 3-Apr-2015.)
 |- 
 <. A ,  B ,  C >.  e.  _V
 
Theoremelop 4111 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 4112 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 4113 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 4114 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 4115 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 4116 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 4117 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 4118 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 4119 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 4120 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
 |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
 ( A  =  C  /\  B  =  D ) )
 
Theoremotth2 4121 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 4122 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 4123* 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 4124* 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 4125* Closed theorem form of copsex2g 4126. (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 4126* 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 4127* 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 4128 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 -.  (/)  e.  <. A ,  B >.
 
Theoremopeqex 4129 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 4130 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 4131 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 4132 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 4133* "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 4134 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 4135 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 4136* "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 4137* "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 4138* 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 4139* 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 4140 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 3533 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 4141 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 4142 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 4143 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 4144* 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 4145* 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 4146* 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 4147* 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 4148* 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 4149* Closed theorem form of opelopab 4158. (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 4150* 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 4151* 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 4152* 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 4153* 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 4154* 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 4155* 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 4156* 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 4157* 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 4158* 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 4159* 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 4160* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4158 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 4161* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4158 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 4162 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 4163 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 4164 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
 |-  ( ph  ->  ps )   =>    |-  { <. x ,  y >.  |  ph } 
 C_  { <. x ,  y >.  |  ps }
 
Theoremssopab2dv 4165* 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 4166 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 4167 Non-empty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.)
 |-  ( { <. x ,  y >.  |  ph }  =/=  (/)  <->  E. x E. y ph )
 
Theoremiunopab 4168* 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 4169 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 4170 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 4171 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 4172 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 4173 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 4174 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 4175 Extend class notation to include the epsilon relation.
 class  _E
 
Syntaxcid 4176 Extend the definition of a class to include identity relation.
 class  _I
 
Definitiondf-eprel 4177* 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 4178. Thus,  5  _E  { 1 ,  5 } (ex-eprel 20451). (Contributed by NM, 13-Aug-1995.)
 |- 
 _E  =  { <. x ,  y >.  |  x  e.  y }
 
Theoremepelg 4178 The epsilon relation and membership are the same. General version of epel 4180. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
 
Theoremepelc 4179 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 )
 
Theoremepel 4180 The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)
 |-  ( x  _E  y  <->  x  e.  y )
 
Definitiondf-id 4181* Define the identity relation. Definition 9.15 of [Quine] p. 64. For example,  5  _I  5 and 
-.  4  _I  5 (ex-id 20452). (Contributed by NM, 13-Aug-1995.)
 |- 
 _I  =  { <. x ,  y >.  |  x  =  y }
 
Theoremdfid3 4182 A stronger version of df-id 4181 that doesn't require  x and  y to be distinct. Ordinarily, we wouldn't use this as a definition, since non-distinct dummy variables would make soundness verification more difficult (as the proof here shows). The proof can be instructive in showing how distinct variable requirements may be eliminated, a task that is not necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by Mario Carneiro, 18-Nov-2016.)
 |- 
 _I  =  { <. x ,  y >.  |  x  =  y }
 
Theoremdfid2 4183 Alternate definition of the identity relation. (Contributed by NM, 15-Mar-2007.)
 |- 
 _I  =  { <. x ,  x >.  |  x  =  x }
 
2.3.7  Partial and complete ordering
 
Syntaxwpo 4184 Extend wff notation to include the strict partial ordering predicate. Read: '  R is a partial order on  A.'
 wff  R  Po  A
 
Syntaxwor 4185 Extend wff notation to include the strict complete ordering predicate. Read: '  R orders  A.'
 wff  R  Or  A
 
Definitiondf-po 4186* Define the strict partial order predicate. Definition of [Enderton] p. 168. The expression  R  Po  A means  R is a partial order on  A. For example,  <  Po  RR is true, while  <_  Po  RR is false (ex-po 20453). (Contributed by NM, 16-Mar-1997.)
 |-  ( R  Po  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
 
Definitiondf-so 4187* Define the strict complete (linear) order predicate. The expression  R  Or  A is true if relationship  R orders  A. For example,  <  Or  RR is true (ltso 8756). (Contributed by NM, 21-Jan-1996.)
 |-  ( R  Or  A  <->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) ) )
 
Theoremposs 4188 Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  ( A  C_  B  ->  ( R  Po  B  ->  R  Po  A ) )
 
Theorempoeq1 4189 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
 
Theorempoeq2 4190 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
 |-  ( A  =  B  ->  ( R  Po  A  <->  R  Po  B ) )
 
Theoremnfpo 4191 Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  Po  A
 
Theoremnfso 4192 Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/ x  R  Or  A
 
Theorempocl 4193 Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
 |-  ( R  Po  A  ->  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) 
 ->  ( -.  B R B  /\  ( ( B R C  /\  C R D )  ->  B R D ) ) ) )
 
Theoremispod 4194* Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  -.  x R x )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  ->  ( ( x R y  /\  y R z )  ->  x R z ) )   =>    |-  ( ph  ->  R  Po  A )
 
Theoremswopolem 4195* Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )
 )  ->  ( x R y  ->  ( x R z  \/  z R y ) ) )   =>    |-  ( ( ph  /\  ( X  e.  A  /\  Y  e.  A  /\  Z  e.  A )
 )  ->  ( X R Y  ->  ( X R Z  \/  Z R Y ) ) )
 
Theoremswopo 4196* A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( ph  /\  (
 y  e.  A  /\  z  e.  A )
 )  ->  ( y R z  ->  -.  z R y ) )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A )
 )  ->  ( x R y  ->  ( x R z  \/  z R y ) ) )   =>    |-  ( ph  ->  R  Po  A )
 
Theorempoirr 4197 A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
 
Theorempotr 4198 A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  ( ( B R C  /\  C R D )  ->  B R D ) )
 
Theorempo2nr 4199 A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A ) )  ->  -.  ( B R C  /\  C R B ) )
 
Theorempo3nr 4200 A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)
 |-  ( ( R  Po  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A ) )  ->  -.  ( B R C  /\  C R D  /\  D R B ) )
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