Theorem prid1g 3622

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 2137	. . 3 |- A = A
2	1	orci 720	. 2 |- ( A = A \/ A = B )
3		elprg 3542	. 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 697   = wceq 1331   e. wcel 1480   {cpr 3523

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529

This theorem is referenced by:  prid2g  3623  prid1  3624  preqr1g  3688  opth1  4153  en2lp  4464  acexmidlemcase  5762  en2eqpr  6794  m1expcl2  10308  maxabslemval  10973  xrmaxiflemval  11012  xrmaxaddlem  11022  2strbasg  12049  2strbas1g  12052  coseq0negpitopi  12906