Theorem prid2g 3727

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 3726	. 2 |- ( B e. V -> B e. { B , A } )
2		prcom 3698	. 2 |- { B , A } = { A , B }
3	1, 2	eleqtrdi 2289	1 |- ( B e. V -> B e. { A , B } )

Syntax hints:   -> wi 4   e. wcel 2167   {cpr 3623

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629

This theorem is referenced by:  en2lp  4590  pw2f1odclem  6895  en2eqpr  6968  maxleim  11370  maxabslemval  11373  xrmaxleim  11409  xrmaxiflemval  11415  xrmaxaddlem  11425  2stropg  12798  2strop1g  12801  coseq0negpitopi  15072