Theorem prid1g 3811

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 2238	. . 3 |- A = A
2	1	orci 743	. 2 |- ( A = A \/ A = B )
3		elprg 3725	. 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 720   = wceq 1402   e. wcel 2209   {cpr 3706

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712

This theorem is referenced by:  prid2g  3812  prid1  3813  preqr1g  3886  opth1  4371  en2lp  4696  acexmidlemcase  6070  pw2f1odclem  7124  en2eqpr  7204  m1expcl2  10976  maxabslemval  11952  xrmaxiflemval  11994  xrmaxaddlem  12004  2strbasg  13451  2strbas1g  13454  coseq0negpitopi  15860  structvtxval  16194  umgrnloopv  16269  umgredgprv  16270  umgrpredgv  16302  uhgr2edg  16361  umgrvad2edg  16366  usgr2v1e2w  16401  1hegrvtxdg1fi  16464  vdegp1bid  16470