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Theorem dfopg 4055
Description: Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
Assertion
Ref Expression
dfopg  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )

Proof of Theorem dfopg
StepHypRef Expression
1 elex 2979 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2979 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 dfopif 4054 . . 3  |-  <. A ,  B >.  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
4 iftrue 3795 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )  =  { { A } ,  { A ,  B } } )
53, 4syl5eq 2485 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
61, 2, 5syl2an 477 1  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2970   (/)c0 3635   ifcif 3789   {csn 3875   {cpr 3877   <.cop 3881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-v 2972  df-dif 3329  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-op 3882
This theorem is referenced by:  dfop  4056  opnz  4561  opth1  4563  opth  4564  0nelop  4579  opwf  8017  rankopb  8057  wunop  8887  tskop  8936  gruop  8970  bj-elopg  32519
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