Theorem prid1g 3635

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

Step	Hyp	Ref	Expression
1		eqid 2140	. . 3 |- A = A
2	1	orci 721	. 2 |- ( A = A \/ A = B )
3		elprg 3552	. 2 |- ( A e. V -> ( A e. { A , B } <-> ( A = A \/ A = B ) ) )
4	2, 3	mpbiri 167	1 |- ( A e. V -> A e. { A , B } )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1332   e. wcel 1481   {cpr 3533

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539

This theorem is referenced by:  prid2g  3636  prid1  3637  preqr1g  3701  opth1  4166  en2lp  4477  acexmidlemcase  5777  en2eqpr  6809  m1expcl2  10346  maxabslemval  11012  xrmaxiflemval  11051  xrmaxaddlem  11061  2strbasg  12099  2strbas1g  12102  coseq0negpitopi  12965