Theorem prid1g 3795

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 2232	. . 3 |- A = A
2	1	orci 739	. 2 |- ( A = A \/ A = B )
3		elprg 3709	. 2 |- ( A e. V -> ( A e. { A , B } <-> ( A = A \/ A = B ) ) )
4	2, 3	mpbiri 168	1 |- ( A e. V -> A e. { A , B } )

Syntax hints:   -> wi 4   \/ wo 716   = wceq 1398   e. wcel 2203   {cpr 3690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696

This theorem is referenced by:  prid2g  3796  prid1  3797  preqr1g  3870  opth1  4352  en2lp  4676  acexmidlemcase  6045  pw2f1odclem  7087  en2eqpr  7167  m1expcl2  10923  maxabslemval  11893  xrmaxiflemval  11935  xrmaxaddlem  11945  2strbasg  13333  2strbas1g  13336  coseq0negpitopi  15701  structvtxval  16034  umgrnloopv  16109  umgredgprv  16110  umgrpredgv  16142  uhgr2edg  16201  umgrvad2edg  16206  usgr2v1e2w  16241  1hegrvtxdg1fi  16304  vdegp1bid  16310