Theorem prid1g 3775

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 2231	. . 3 |- A = A
2	1	orci 738	. 2 |- ( A = A \/ A = B )
3		elprg 3689	. 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 715   = wceq 1397   e. wcel 2202   {cpr 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  prid2g  3776  prid1  3777  preqr1g  3849  opth1  4328  en2lp  4652  acexmidlemcase  6012  pw2f1odclem  7019  en2eqpr  7098  m1expcl2  10822  maxabslemval  11768  xrmaxiflemval  11810  xrmaxaddlem  11820  2strbasg  13202  2strbas1g  13205  coseq0negpitopi  15559  structvtxval  15889  umgrnloopv  15964  umgredgprv  15965  umgrpredgv  15997  uhgr2edg  16056  umgrvad2edg  16061  usgr2v1e2w  16096  1hegrvtxdg1fi  16159  vdegp1bid  16165