Theorem prid2g 4690

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 4689	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵, 𝐴})
2		prcom 4661	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
3	1, 2	eleqtrdi 2923	1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∈ wcel 2110  {cpr 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-un 3940  df-sn 4561  df-pr 4563

This theorem is referenced by:  prel12g  4787  prproe  4829  unisn2  5208  fr2nr  5527  fpr2g  6968  f1prex  7034  pw2f1olem  8615  hashprdifel  13753  gcdcllem3  15844  mgm2nsgrplem1  18077  mgm2nsgrplem2  18078  mgm2nsgrplem3  18079  sgrp2nmndlem1  18082  sgrp2rid2  18085  pmtrprfv  18575  m2detleib  21234  indistopon  21603  pptbas  21610  coseq0negpitopi  25083  uhgr2edg  26984  umgrvad2edg  26989  uspgr2v1e2w  27027  usgr2v1e2w  27028  nb3grprlem1  27156  nb3grprlem2  27157  1hegrvtxdg1  27283  cyc3genpmlem  30788  prsiga  31385  bj-prmoore  34401  ftc1anclem8  34968  pr2el2  39903  pr2eldif2  39907  fourierdlem54  42439  prsal  42597  sge0pr  42670  imarnf1pr  43475  paireqne  43667  1hegrlfgr  44001