Theorem prid1g 3737

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 2205	. . 3 |- A = A
2	1	orci 733	. 2 |- ( A = A \/ A = B )
3		elprg 3653	. 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 710   = wceq 1373   e. wcel 2176   {cpr 3634

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640

This theorem is referenced by:  prid2g  3738  prid1  3739  preqr1g  3807  opth1  4280  en2lp  4602  acexmidlemcase  5939  pw2f1odclem  6931  en2eqpr  7004  m1expcl2  10706  maxabslemval  11519  xrmaxiflemval  11561  xrmaxaddlem  11571  2strbasg  12952  2strbas1g  12955  coseq0negpitopi  15308  structvtxval  15636