Theorem prid2 3630

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 3629	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3599	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2214	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 1480  Vcvv 2686  {cpr 3528

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534

This theorem is referenced by:  prel12  3698  opi2  4155  opeluu  4371  ontr2exmid  4440  onsucelsucexmid  4445  regexmidlemm  4447  ordtri2or2exmid  4486  dmrnssfld  4802  funopg  5157  acexmidlema  5765  acexmidlemcase  5769  acexmidlem2  5771  1lt2o  6339  2dom  6699  unfiexmid  6806  djuss  6955  exmidonfinlem  7049  exmidfodomrlemr  7058  exmidfodomrlemrALT  7059  exmidaclem  7064  cnelprrecn  7756  mnfxr  7822  sup3exmid  8715  m1expcl2  10315  bdop  13073  isomninnlem  13225