Theorem prid2g 3771

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 3770	. 2 |- ( B e. V -> B e. { B , A } )
2		prcom 3742	. 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 3667

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 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  en2lp  4646  pw2f1odclem  7003  en2eqpr  7080  maxleim  11731  maxabslemval  11734  xrmaxleim  11770  xrmaxiflemval  11776  xrmaxaddlem  11786  2stropg  13169  2strop1g  13172  coseq0negpitopi  15525  umgredgprv  15930  umgrpredgv  15960  uhgr2edg  16019  umgrvad2edg  16024