Theorem prid2 3778

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 3777	. 2 ⊢ 𝐵 ∈ {𝐵, 𝐴}
3		prcom 3747	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
4	2, 3	eleqtri 2306	1 ⊢ 𝐵 ∈ {𝐴, 𝐵}

Syntax hints:   ∈ wcel 2202  Vcvv 2802  {cpr 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  prel12  3854  opi2  4325  opeluu  4547  ontr2exmid  4623  onsucelsucexmid  4628  regexmidlemm  4630  ordtri2or2exmid  4669  ontri2orexmidim  4670  dmrnssfld  4995  funopg  5360  acexmidlema  6008  acexmidlemcase  6012  acexmidlem2  6014  1lt2o  6609  2dom  6979  en2m  6998  unfiexmid  7109  djuss  7268  pr2cv1  7399  exmidonfinlem  7403  exmidfodomrlemr  7412  exmidfodomrlemrALT  7413  exmidaclem  7422  cnelprrecn  8167  mnfxr  8235  sup3exmid  9136  m1expcl2  10822  fun2dmnop0  11110  fnpr2ob  13422  lgsdir2lem3  15758  upgrex  15953  upgr1een  15974  bdop  16470  2o01f  16593  iswomni0  16655