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Theorem dfop 3811
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 3810 . 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 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   {cpr 3654   <.cop 3656
This theorem is referenced by:  opid  3830  elop  4255  opi1  4256  opi2  4257  opeqsn  4278  opeqpr  4279  uniop  4285  op1stb  4585  xpsspw  4813  xpsspwOLD  4814  relop  4850  funopg  5302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-op 3662
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