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Theorem dfop 3926
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 3925 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
41, 2, 3mp2an 654 1  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   _Vcvv 2900   {csn 3758   {cpr 3759   <.cop 3761
This theorem is referenced by:  opid  3945  elop  4371  opi1  4372  opi2  4373  opeqsn  4394  opeqpr  4395  uniop  4401  op1stb  4699  xpsspw  4927  xpsspwOLD  4928  relop  4964  funopg  5426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-dif 3267  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-op 3767
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