Theorem prid2g 3594

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 3593	. 2 |- ( B e. V -> B e. { B , A } )
2		prcom 3565	. 2 |- { B , A } = { A , B }
3	1, 2	syl6eleq 2207	1 |- ( B e. V -> B e. { A , B } )

Syntax hints:   -> wi 4   e. wcel 1463   {cpr 3494

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500

This theorem is referenced by:  en2lp  4429  en2eqpr  6754  maxleim  10869  maxabslemval  10872  xrmaxleim  10905  xrmaxiflemval  10911  xrmaxaddlem  10921  2stropg  11904  2strop1g  11907