Theorem prid2 3730

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 3729	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3699	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2271	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2167  Vcvv 2763  {cpr 3624

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 3629  df-pr 3630

This theorem is referenced by:  prel12  3802  opi2  4267  opeluu  4486  ontr2exmid  4562  onsucelsucexmid  4567  regexmidlemm  4569  ordtri2or2exmid  4608  ontri2orexmidim  4609  dmrnssfld  4930  funopg  5293  acexmidlema  5916  acexmidlemcase  5920  acexmidlem2  5922  1lt2o  6509  2dom  6873  unfiexmid  6988  djuss  7145  exmidonfinlem  7272  exmidfodomrlemr  7281  exmidfodomrlemrALT  7282  exmidaclem  7291  cnelprrecn  8032  mnfxr  8100  sup3exmid  9001  m1expcl2  10670  fnpr2ob  13042  lgsdir2lem3  15355  bdop  15605  2o01f  15725  iswomni0  15782