Theorem prid2g 3724

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 3723	. 2 |- ( B e. V -> B e. { B , A } )
2		prcom 3695	. 2 |- { B , A } = { A , B }
3	1, 2	eleqtrdi 2286	1 |- ( B e. V -> B e. { A , B } )

Syntax hints:   -> wi 4   e. wcel 2164   {cpr 3620

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626

This theorem is referenced by:  en2lp  4587  pw2f1odclem  6892  en2eqpr  6965  maxleim  11352  maxabslemval  11355  xrmaxleim  11390  xrmaxiflemval  11396  xrmaxaddlem  11406  2stropg  12741  2strop1g  12744  coseq0negpitopi  15012