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Theorem dfopg 3763
Description: Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dfopg  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )

Proof of Theorem dfopg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2741 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2741 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 df-3an 975 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  <->  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) )
43baibr 915 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( x  e.  { { A } ,  { A ,  B } } 
<->  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) ) )
54abbidv 2288 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  x  e.  { { A } ,  { A ,  B } } }  =  { x  |  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) } )
6 abid2 2291 . . . 4  |-  { x  |  x  e.  { { A } ,  { A ,  B } } }  =  { { A } ,  { A ,  B } }
7 df-op 3592 . . . . 5  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
87eqcomi 2174 . . . 4  |-  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  <. A ,  B >.
95, 6, 83eqtr3g 2226 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =  <. A ,  B >. )
109eqcomd 2176 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
111, 2, 10syl2an 287 1  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 973    = wceq 1348    e. wcel 2141   {cab 2156   _Vcvv 2730   {csn 3583   {cpr 3584   <.cop 3586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-op 3592
This theorem is referenced by:  dfop  3764  opexg  4213  opth1  4221  opth  4222  0nelop  4233  op1stbg  4464
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