Theorem prid1g 3687

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 2170	. . 3 |- A = A
2	1	orci 726	. 2 |- ( A = A \/ A = B )
3		elprg 3603	. 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 703   = wceq 1348   e. wcel 2141   {cpr 3584

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590

This theorem is referenced by:  prid2g  3688  prid1  3689  preqr1g  3753  opth1  4221  en2lp  4538  acexmidlemcase  5848  en2eqpr  6885  m1expcl2  10498  maxabslemval  11172  xrmaxiflemval  11213  xrmaxaddlem  11223  2strbasg  12519  2strbas1g  12522  coseq0negpitopi  13551