Theorem prid2 3729

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 3728	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3698	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2271	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2167  Vcvv 2763  {cpr 3623

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629

This theorem is referenced by:  prel12  3801  opi2  4266  opeluu  4485  ontr2exmid  4561  onsucelsucexmid  4566  regexmidlemm  4568  ordtri2or2exmid  4607  ontri2orexmidim  4608  dmrnssfld  4929  funopg  5292  acexmidlema  5913  acexmidlemcase  5917  acexmidlem2  5919  1lt2o  6500  2dom  6864  unfiexmid  6979  djuss  7136  exmidonfinlem  7260  exmidfodomrlemr  7269  exmidfodomrlemrALT  7270  exmidaclem  7275  cnelprrecn  8015  mnfxr  8083  sup3exmid  8984  m1expcl2  10653  fnpr2ob  12983  lgsdir2lem3  15271  bdop  15521  2o01f  15641  iswomni0  15695