Theorem prid1g 3747

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 2207	. . 3 |- A = A
2	1	orci 733	. 2 |- ( A = A \/ A = B )
3		elprg 3663	. 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 2178   {cpr 3644

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650

This theorem is referenced by:  prid2g  3748  prid1  3749  preqr1g  3820  opth1  4298  en2lp  4620  acexmidlemcase  5962  pw2f1odclem  6956  en2eqpr  7030  m1expcl2  10743  maxabslemval  11634  xrmaxiflemval  11676  xrmaxaddlem  11686  2strbasg  13067  2strbas1g  13070  coseq0negpitopi  15423  structvtxval  15753  umgrnloopvv  15825  umgrpredgv  15851