Theorem prid1g 3474

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 2040	. . 3 |- A = A
2	1	orci 650	. 2 |- ( A = A \/ A = B )
3		elprg 3395	. 2 |- ( A e. V -> ( A e. { A , B } <-> ( A = A \/ A = B ) ) )
4	2, 3	mpbiri 157	1 |- ( A e. V -> A e. { A , B } )

Syntax hints:   -> wi 4   \/ wo 629   = wceq 1243   e. wcel 1393   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by:  prid2g  3475  prid1  3476  preqr1g  3537  opth1  3973  en2lp  4278  acexmidlemcase  5507  m1expcl2  9277