Theorem prid1g 3736

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 2204	. . 3 |- A = A
2	1	orci 732	. 2 |- ( A = A \/ A = B )
3		elprg 3652	. 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 709   = wceq 1372   e. wcel 2175   {cpr 3633

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639

This theorem is referenced by:  prid2g  3737  prid1  3738  preqr1g  3806  opth1  4279  en2lp  4601  acexmidlemcase  5938  pw2f1odclem  6930  en2eqpr  7003  m1expcl2  10704  maxabslemval  11490  xrmaxiflemval  11532  xrmaxaddlem  11542  2strbasg  12923  2strbas1g  12926  coseq0negpitopi  15279  structvtxval  15607