Theorem prid2g 3688

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 3687	. 2 |- ( B e. V -> B e. { B , A } )
2		prcom 3659	. 2 |- { B , A } = { A , B }
3	1, 2	eleqtrdi 2263	1 |- ( B e. V -> B e. { A , B } )

Syntax hints:   -> wi 4   e. wcel 2141   {cpr 3584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590

This theorem is referenced by:  en2lp  4538  en2eqpr  6885  maxleim  11169  maxabslemval  11172  xrmaxleim  11207  xrmaxiflemval  11213  xrmaxaddlem  11223  2stropg  12520  2strop1g  12523  coseq0negpitopi  13551