Theorem prid2g 3774

Description: An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Assertion

Ref	Expression
prid2g	|- ( B e. V -> B e. { A , B } )

Proof of Theorem prid2g

Step	Hyp	Ref	Expression
1		prid1g 3773	. 2 |- ( B e. V -> B e. { B , A } )
2		prcom 3745	. 2 |- { B , A } = { A , B }
3	1, 2	eleqtrdi 2322	1 |- ( B e. V -> B e. { A , B } )

Syntax hints:   -> wi 4   e. wcel 2200   {cpr 3668

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674

This theorem is referenced by:  en2lp  4650  pw2f1odclem  7015  en2eqpr  7092  maxleim  11756  maxabslemval  11759  xrmaxleim  11795  xrmaxiflemval  11801  xrmaxaddlem  11811  2stropg  13194  2strop1g  13197  coseq0negpitopi  15550  umgredgprv  15956  umgrpredgv  15986  uhgr2edg  16045  umgrvad2edg  16050  usgr2v1e2w  16085