Theorem prid1g 3800

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 2234	. . 3 |- A = A
2	1	orci 739	. 2 |- ( A = A \/ A = B )
3		elprg 3714	. 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 2205   {cpr 3695

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701

This theorem is referenced by:  prid2g  3801  prid1  3802  preqr1g  3875  opth1  4357  en2lp  4681  acexmidlemcase  6053  pw2f1odclem  7100  en2eqpr  7180  m1expcl2  10947  maxabslemval  11918  xrmaxiflemval  11960  xrmaxaddlem  11970  2strbasg  13417  2strbas1g  13420  coseq0negpitopi  15827  structvtxval  16160  umgrnloopv  16235  umgredgprv  16236  umgrpredgv  16268  uhgr2edg  16327  umgrvad2edg  16332  usgr2v1e2w  16367  1hegrvtxdg1fi  16430  vdegp1bid  16436