Theorem prid2 3683

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 3682	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3652	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2241	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2136  Vcvv 2726  {cpr 3577

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  prel12  3751  opi2  4211  opeluu  4428  ontr2exmid  4502  onsucelsucexmid  4507  regexmidlemm  4509  ordtri2or2exmid  4548  ontri2orexmidim  4549  dmrnssfld  4867  funopg  5222  acexmidlema  5833  acexmidlemcase  5837  acexmidlem2  5839  1lt2o  6410  2dom  6771  unfiexmid  6883  djuss  7035  exmidonfinlem  7149  exmidfodomrlemr  7158  exmidfodomrlemrALT  7159  exmidaclem  7164  cnelprrecn  7889  mnfxr  7955  sup3exmid  8852  m1expcl2  10477  lgsdir2lem3  13571  bdop  13757  2o01f  13876  iswomni0  13930