Theorem prid2 3774

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 3773	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3743	. 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  3850  opi2  4321  opeluu  4543  ontr2exmid  4619  onsucelsucexmid  4624  regexmidlemm  4626  ordtri2or2exmid  4665  ontri2orexmidim  4666  dmrnssfld  4991  funopg  5356  acexmidlema  6002  acexmidlemcase  6006  acexmidlem2  6008  1lt2o  6603  2dom  6973  en2m  6992  unfiexmid  7101  djuss  7258  pr2cv1  7389  exmidonfinlem  7392  exmidfodomrlemr  7401  exmidfodomrlemrALT  7402  exmidaclem  7411  cnelprrecn  8156  mnfxr  8224  sup3exmid  9125  m1expcl2  10811  fun2dmnop0  11098  fnpr2ob  13410  lgsdir2lem3  15746  upgrex  15940  bdop  16380  2o01f  16503  iswomni0  16565