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Theorem dfop 4202
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.) (Avoid depending on this detail.)
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 4201 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
41, 2, 3mp2an 670 1  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823   _Vcvv 3106   {csn 4016   {cpr 4018   <.cop 4022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-op 4023
This theorem is referenced by:  opid  4222  elop  4703  opi1  4704  opi2  4705  op1stb  4707  opeqsn  4732  opeqpr  4733  uniop  4739  xpsspw  5104  xpsspwOLD  5105  relop  5142  funopg  5602
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