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Theorem prid1g 3502
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 2056 . . 3  |-  A  =  A
21orci 660 . 2  |-  ( A  =  A  \/  A  =  B )
3 elprg 3423 . 2  |-  ( A  e.  V  ->  ( A  e.  { A ,  B }  <->  ( A  =  A  \/  A  =  B ) ) )
42, 3mpbiri 161 1  |-  ( A  e.  V  ->  A  e.  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 639    = wceq 1259    e. wcel 1409   {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410
This theorem is referenced by:  prid2g  3503  prid1  3504  preqr1g  3565  opth1  4001  en2lp  4306  acexmidlemcase  5535  m1expcl2  9442
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