Theorem prid2 3776

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 3775	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3745	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2304	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2200  Vcvv 2800  {cpr 3668

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 2802  df-un 3202  df-sn 3673  df-pr 3674

This theorem is referenced by:  prel12  3852  opi2  4323  opeluu  4545  ontr2exmid  4621  onsucelsucexmid  4626  regexmidlemm  4628  ordtri2or2exmid  4667  ontri2orexmidim  4668  dmrnssfld  4993  funopg  5358  acexmidlema  6004  acexmidlemcase  6008  acexmidlem2  6010  1lt2o  6605  2dom  6975  en2m  6994  unfiexmid  7103  djuss  7260  pr2cv1  7391  exmidonfinlem  7394  exmidfodomrlemr  7403  exmidfodomrlemrALT  7404  exmidaclem  7413  cnelprrecn  8158  mnfxr  8226  sup3exmid  9127  m1expcl2  10813  fun2dmnop0  11101  fnpr2ob  13413  lgsdir2lem3  15749  upgrex  15944  bdop  16406  2o01f  16529  iswomni0  16591