Theorem prid2g 4766

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 4765	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵, 𝐴})
2		prcom 4737	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
3	1, 2	eleqtrdi 2844	1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∈ wcel 2107  {cpr 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-sn 4630  df-pr 4632

This theorem is referenced by:  prel12g  4865  prproe  4907  unisn2  5313  fr2nr  5655  fpr2g  7213  f1prex  7282  pw2f1olem  9076  hashprdifel  14358  gcdcllem3  16442  mgm2nsgrplem1  18799  mgm2nsgrplem2  18800  mgm2nsgrplem3  18801  sgrp2nmndlem1  18804  sgrp2rid2  18807  pmtrprfv  19321  m2detleib  22133  indistopon  22504  pptbas  22511  coseq0negpitopi  26013  uhgr2edg  28496  umgrvad2edg  28501  uspgr2v1e2w  28539  usgr2v1e2w  28540  nb3grprlem1  28668  nb3grprlem2  28669  1hegrvtxdg1  28795  cyc3genpmlem  32341  elrspunsn  32578  prsiga  33160  bj-prmoore  36044  ftc1anclem8  36616  pr2el2  42350  pr2eldif2  42354  fourierdlem54  44924  prsal  45082  sge0pr  45158  imarnf1pr  46038  paireqne  46227  1hegrlfgr  46558  lubprlem  47643