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Theorem dfopg 3739
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 2723 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2723 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 df-3an 965 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  <->  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) )
43baibr 906 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( x  e.  { { A } ,  { A ,  B } } 
<->  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) ) )
54abbidv 2275 . . . 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 2278 . . . 4  |-  { x  |  x  e.  { { A } ,  { A ,  B } } }  =  { { A } ,  { A ,  B } }
7 df-op 3569 . . . . 5  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
87eqcomi 2161 . . . 4  |-  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  <. A ,  B >.
95, 6, 83eqtr3g 2213 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =  <. A ,  B >. )
109eqcomd 2163 . 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 963    = wceq 1335    e. wcel 2128   {cab 2143   _Vcvv 2712   {csn 3560   {cpr 3561   <.cop 3563
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714  df-op 3569
This theorem is referenced by:  dfop  3740  opexg  4188  opth1  4196  opth  4197  0nelop  4208  op1stbg  4439
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