Theorem prid2 3803

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 3802	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3772	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2309	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2205  Vcvv 2815  {cpr 3695

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701

This theorem is referenced by:  prel12  3880  opi2  4354  opeluu  4576  ontr2exmid  4652  onsucelsucexmid  4657  regexmidlemm  4659  ordtri2or2exmid  4698  ontri2orexmidim  4699  dmrnssfld  5025  funopg  5391  acexmidlema  6049  acexmidlemcase  6053  acexmidlem2  6055  1lt2o  6688  2dom  7059  en2m  7079  unfiexmid  7191  djuss  7374  pr2cv1  7505  exmidonfinlem  7509  exmidfodomrlemr  7518  exmidfodomrlemrALT  7519  exmidaclem  7528  cnelprrecn  8279  mnfxr  8346  sup3exmid  9248  m1expcl2  10947  fun2dmnop0  11247  fnpr2ob  13604  lgsdir2lem3  16029  upgrex  16224  upgr1een  16245  eulerpathprum  16601  bdop  16771  2o01f  16894  iswomni0  16962