Theorem prid2 3700

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 3699	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3669	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2252	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2148  Vcvv 2738  {cpr 3594

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600

This theorem is referenced by:  prel12  3772  opi2  4234  opeluu  4451  ontr2exmid  4525  onsucelsucexmid  4530  regexmidlemm  4532  ordtri2or2exmid  4571  ontri2orexmidim  4572  dmrnssfld  4891  funopg  5251  acexmidlema  5866  acexmidlemcase  5870  acexmidlem2  5872  1lt2o  6443  2dom  6805  unfiexmid  6917  djuss  7069  exmidonfinlem  7192  exmidfodomrlemr  7201  exmidfodomrlemrALT  7202  exmidaclem  7207  cnelprrecn  7947  mnfxr  8014  sup3exmid  8914  m1expcl2  10542  fnpr2ob  12759  lgsdir2lem3  14434  bdop  14630  2o01f  14749  iswomni0  14802