Theorem prid1g 3663

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 2157	. . 3 |- A = A
2	1	orci 721	. 2 |- ( A = A \/ A = B )
3		elprg 3580	. 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 1335   e. wcel 2128   {cpr 3561

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567

This theorem is referenced by:  prid2g  3664  prid1  3665  preqr1g  3729  opth1  4195  en2lp  4511  acexmidlemcase  5813  en2eqpr  6845  m1expcl2  10423  maxabslemval  11090  xrmaxiflemval  11129  xrmaxaddlem  11139  2strbasg  12251  2strbas1g  12254  coseq0negpitopi  13117