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Theorem dfopg 3615
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 2630 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2630 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 df-3an 926 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  <->  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) )
43baibr 867 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( x  e.  { { A } ,  { A ,  B } } 
<->  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) ) )
54abbidv 2205 . . . 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 2208 . . . 4  |-  { x  |  x  e.  { { A } ,  { A ,  B } } }  =  { { A } ,  { A ,  B } }
7 df-op 3450 . . . . 5  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
87eqcomi 2092 . . . 4  |-  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  <. A ,  B >.
95, 6, 83eqtr3g 2143 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =  <. A ,  B >. )
109eqcomd 2093 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
111, 2, 10syl2an 283 1  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   _Vcvv 2619   {csn 3441   {cpr 3442   <.cop 3444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-op 3450
This theorem is referenced by:  dfop  3616  opexg  4046  opth1  4054  opth  4055  0nelop  4066  op1stbg  4291
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