Theorem prid2g 4697

Description: An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Assertion

Ref	Expression
prid2g	⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵})

Proof of Theorem prid2g

Step	Hyp	Ref	Expression
1		prid1g 4696	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵, 𝐴})
2		prcom 4668	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
3	1, 2	eleqtrdi 2923	1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∈ wcel 2114  {cpr 4569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-un 3941  df-sn 4568  df-pr 4570

This theorem is referenced by:  prel12g  4794  prproe  4836  unisn2  5216  fr2nr  5533  fpr2g  6974  f1prex  7040  pw2f1olem  8621  hashprdifel  13760  gcdcllem3  15850  mgm2nsgrplem1  18083  mgm2nsgrplem2  18084  mgm2nsgrplem3  18085  sgrp2nmndlem1  18088  sgrp2rid2  18091  pmtrprfv  18581  m2detleib  21240  indistopon  21609  pptbas  21616  coseq0negpitopi  25089  uhgr2edg  26990  umgrvad2edg  26995  uspgr2v1e2w  27033  usgr2v1e2w  27034  nb3grprlem1  27162  nb3grprlem2  27163  1hegrvtxdg1  27289  cyc3genpmlem  30793  prsiga  31390  bj-prmoore  34410  ftc1anclem8  34989  pr2el2  39930  pr2eldif2  39934  fourierdlem54  42465  prsal  42623  sge0pr  42696  imarnf1pr  43501  paireqne  43693  1hegrlfgr  44027