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Theorem dfopg 4217
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 3127 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 3127 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 dfopif 4216 . . 3  |-  <. A ,  B >.  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
4 iftrue 3951 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )  =  { { A } ,  { A ,  B } } )
53, 4syl5eq 2520 . 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 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   ifcif 3945   {csn 4033   {cpr 4035   <.cop 4039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-dif 3484  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-op 4040
This theorem is referenced by:  dfop  4218  opnz  4724  opth1  4726  opth  4727  0nelop  4743  opwf  8242  rankopb  8282  wunop  9112  tskop  9161  gruop  9195  bj-elopg  34075
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