Theorem prid1g 3773

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 2229	. . 3 |- A = A
2	1	orci 736	. 2 |- ( A = A \/ A = B )
3		elprg 3687	. 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 713   = wceq 1395   e. wcel 2200   {cpr 3668

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674

This theorem is referenced by:  prid2g  3774  prid1  3775  preqr1g  3847  opth1  4326  en2lp  4650  acexmidlemcase  6008  pw2f1odclem  7015  en2eqpr  7092  m1expcl2  10813  maxabslemval  11759  xrmaxiflemval  11801  xrmaxaddlem  11811  2strbasg  13193  2strbas1g  13196  coseq0negpitopi  15550  structvtxval  15880  umgrnloopv  15955  umgredgprv  15956  umgrpredgv  15986  uhgr2edg  16045  umgrvad2edg  16050  usgr2v1e2w  16085