Theorem prid2g 3697

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 3696	. 2 |- ( B e. V -> B e. { B , A } )
2		prcom 3668	. 2 |- { B , A } = { A , B }
3	1, 2	eleqtrdi 2270	1 |- ( B e. V -> B e. { A , B } )

Syntax hints:   -> wi 4   e. wcel 2148   {cpr 3593

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599

This theorem is referenced by:  en2lp  4552  en2eqpr  6904  maxleim  11207  maxabslemval  11210  xrmaxleim  11245  xrmaxiflemval  11251  xrmaxaddlem  11261  2stropg  12571  2strop1g  12574  coseq0negpitopi  14128