Theorem prid2 3517

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 3516	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3486	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2157	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 1434  Vcvv 2610  {cpr 3417

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423

This theorem is referenced by:  prel12  3583  opi2  4016  opeluu  4228  ontr2exmid  4296  onsucelsucexmid  4301  regexmidlemm  4303  ordtri2or2exmid  4342  dmrnssfld  4643  funopg  4984  acexmidlema  5555  acexmidlemcase  5559  acexmidlem2  5561  2dom  6374  unfiexmid  6463  djuss  6564  cnelprrecn  7241  mnfxr  7307  m1expcl2  9665  bdop  10951