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Theorem dfopg 3673
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 2671 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2671 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 df-3an 949 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  <->  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) )
43baibr 890 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( x  e.  { { A } ,  { A ,  B } } 
<->  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) ) )
54abbidv 2235 . . . 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 2238 . . . 4  |-  { x  |  x  e.  { { A } ,  { A ,  B } } }  =  { { A } ,  { A ,  B } }
7 df-op 3506 . . . . 5  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
87eqcomi 2121 . . . 4  |-  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  <. A ,  B >.
95, 6, 83eqtr3g 2173 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =  <. A ,  B >. )
109eqcomd 2123 . 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 947    = wceq 1316    e. wcel 1465   {cab 2103   _Vcvv 2660   {csn 3497   {cpr 3498   <.cop 3500
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662  df-op 3506
This theorem is referenced by:  dfop  3674  opexg  4120  opth1  4128  opth  4129  0nelop  4140  op1stbg  4370
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