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 ∈ V

Assertion

Ref	Expression
prid2	⊢ 𝐵 ∈ {𝐴, 𝐵}

Proof of Theorem prid2

Step	Hyp	Ref	Expression
1		prid2.1	. . 3 ⊢ 𝐵 ∈ V
2	1	prid1 3772	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3742	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2304	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ 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  3848  opi2  4318  opeluu  4540  ontr2exmid  4616  onsucelsucexmid  4621  regexmidlemm  4623  ordtri2or2exmid  4662  ontri2orexmidim  4663  dmrnssfld  4986  funopg  5351  acexmidlema  5991  acexmidlemcase  5995  acexmidlem2  5997  1lt2o  6586  2dom  6956  en2m  6972  unfiexmid  7076  djuss  7233  pr2cv1  7364  exmidonfinlem  7367  exmidfodomrlemr  7376  exmidfodomrlemrALT  7377  exmidaclem  7386  cnelprrecn  8131  mnfxr  8199  sup3exmid  9100  m1expcl2  10778  fun2dmnop0  11064  fnpr2ob  13368  lgsdir2lem3  15703  upgrex  15897  bdop  16196  2o01f  16317  iswomni0  16378