Theorem prid2 3798

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 3797	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3767	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2307	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2203  Vcvv 2813  {cpr 3690

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696

This theorem is referenced by:  prel12  3875  opi2  4349  opeluu  4571  ontr2exmid  4647  onsucelsucexmid  4652  regexmidlemm  4654  ordtri2or2exmid  4693  ontri2orexmidim  4694  dmrnssfld  5020  funopg  5386  acexmidlema  6041  acexmidlemcase  6045  acexmidlem2  6047  1lt2o  6675  2dom  7046  en2m  7066  unfiexmid  7178  djuss  7361  pr2cv1  7492  exmidonfinlem  7496  exmidfodomrlemr  7505  exmidfodomrlemrALT  7506  exmidaclem  7515  cnelprrecn  8263  mnfxr  8330  sup3exmid  9231  m1expcl2  10923  fun2dmnop0  11222  fnpr2ob  13553  lgsdir2lem3  15903  upgrex  16098  upgr1een  16119  eulerpathprum  16475  bdop  16645  2o01f  16768  iswomni0  16836