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Theorem prid1g 3595
Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
Assertion
Ref Expression
prid1g  |-  ( A  e.  V  ->  A  e.  { A ,  B } )

Proof of Theorem prid1g
StepHypRef Expression
1 eqid 2115 . . 3  |-  A  =  A
21orci 703 . 2  |-  ( A  =  A  \/  A  =  B )
3 elprg 3515 . 2  |-  ( A  e.  V  ->  ( A  e.  { A ,  B }  <->  ( A  =  A  \/  A  =  B ) ) )
42, 3mpbiri 167 1  |-  ( A  e.  V  ->  A  e.  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 680    = wceq 1314    e. wcel 1463   {cpr 3496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502
This theorem is referenced by:  prid2g  3596  prid1  3597  preqr1g  3661  opth1  4126  en2lp  4437  acexmidlemcase  5735  en2eqpr  6767  m1expcl2  10255  maxabslemval  10920  xrmaxiflemval  10959  xrmaxaddlem  10969  2strbasg  11955  2strbas1g  11958
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