Theorem prid2 3701

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 3700	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3670	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2252	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2148  Vcvv 2739  {cpr 3595

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 2741  df-un 3135  df-sn 3600  df-pr 3601

This theorem is referenced by:  prel12  3773  opi2  4235  opeluu  4452  ontr2exmid  4526  onsucelsucexmid  4531  regexmidlemm  4533  ordtri2or2exmid  4572  ontri2orexmidim  4573  dmrnssfld  4892  funopg  5252  acexmidlema  5869  acexmidlemcase  5873  acexmidlem2  5875  1lt2o  6446  2dom  6808  unfiexmid  6920  djuss  7072  exmidonfinlem  7195  exmidfodomrlemr  7204  exmidfodomrlemrALT  7205  exmidaclem  7210  cnelprrecn  7950  mnfxr  8017  sup3exmid  8917  m1expcl2  10545  fnpr2ob  12766  lgsdir2lem3  14592  bdop  14788  2o01f  14908  iswomni0  14961