Theorem prid2g 3743

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 3742	. 2 |- ( B e. V -> B e. { B , A } )
2		prcom 3714	. 2 |- { B , A } = { A , B }
3	1, 2	eleqtrdi 2299	1 |- ( B e. V -> B e. { A , B } )

Syntax hints:   -> wi 4   e. wcel 2177   {cpr 3639

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645

This theorem is referenced by:  en2lp  4615  pw2f1odclem  6951  en2eqpr  7025  maxleim  11601  maxabslemval  11604  xrmaxleim  11640  xrmaxiflemval  11646  xrmaxaddlem  11656  2stropg  13038  2strop1g  13041  coseq0negpitopi  15393  umgrpredgv  15821