Theorem prid2 3773

Description: An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
prid2.1	|- B e. _V

Assertion

Ref	Expression
prid2	|- B e. { A , B }

Proof of Theorem prid2

Step	Hyp	Ref	Expression
1		prid2.1	. . 3 |- B e. _V
2	1	prid1 3772	. 2 |- B e. { B , A }
3		prcom 3742	. 2 |- { B , A } = { A , B }
4	2, 3	eleqtri 2304	1 |- B e. { A , B }

Syntax hints:   e. wcel 2200   _Vcvv 2799   {cpr 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  prel12  3849  opi2  4320  opeluu  4542  ontr2exmid  4618  onsucelsucexmid  4623  regexmidlemm  4625  ordtri2or2exmid  4664  ontri2orexmidim  4665  dmrnssfld  4990  funopg  5355  acexmidlema  6001  acexmidlemcase  6005  acexmidlem2  6007  1lt2o  6601  2dom  6971  en2m  6987  unfiexmid  7096  djuss  7253  pr2cv1  7384  exmidonfinlem  7387  exmidfodomrlemr  7396  exmidfodomrlemrALT  7397  exmidaclem  7406  cnelprrecn  8151  mnfxr  8219  sup3exmid  9120  m1expcl2  10800  fun2dmnop0  11087  fnpr2ob  13394  lgsdir2lem3  15730  upgrex  15924  bdop  16347  2o01f  16471  iswomni0  16533