Theorem prid2 3666

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 3665	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3635	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2232	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2128  Vcvv 2712  {cpr 3561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567

This theorem is referenced by:  prel12  3734  opi2  4193  opeluu  4410  ontr2exmid  4484  onsucelsucexmid  4489  regexmidlemm  4491  ordtri2or2exmid  4530  ontri2orexmidim  4531  dmrnssfld  4849  funopg  5204  acexmidlema  5815  acexmidlemcase  5819  acexmidlem2  5821  1lt2o  6389  2dom  6750  unfiexmid  6862  djuss  7014  exmidonfinlem  7128  exmidfodomrlemr  7137  exmidfodomrlemrALT  7138  exmidaclem  7143  cnelprrecn  7868  mnfxr  7934  sup3exmid  8828  m1expcl2  10441  bdop  13461  2o01f  13579  iswomni0  13633