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Theorem dfop 3795
Description: Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.)
Hypotheses
Ref Expression
dfop.1  |-  A  e. 
_V
dfop.2  |-  B  e. 
_V
Assertion
Ref Expression
dfop  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }

Proof of Theorem dfop
StepHypRef Expression
1 dfop.1 . 2  |-  A  e. 
_V
2 dfop.2 . 2  |-  B  e. 
_V
3 dfopg 3794 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
41, 2, 3mp2an 653 1  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   {cpr 3641   <.cop 3643
This theorem is referenced by:  opid  3814  elop  4239  opi1  4240  opi2  4241  opeqsn  4262  opeqpr  4263  uniop  4269  op1stb  4569  xpsspw  4797  xpsspwOLD  4798  relop  4834  funopg  5286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-op 3649
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