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 2849	1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∈ wcel 2106  {cpr 4633

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634

This theorem is referenced by:  prel12g  4869  prproe  4910  unisn2  5318  fr2nr  5666  fpr2g  7231  f1prex  7304  fvf1pr  7327  pw2f1olem  9115  hashprdifel  14434  gcdcllem3  16535  mgm2nsgrplem1  18944  mgm2nsgrplem2  18945  mgm2nsgrplem3  18946  sgrp2nmndlem1  18949  sgrp2rid2  18952  pmtrprfv  19486  m2detleib  22653  indistopon  23024  pptbas  23031  coseq0negpitopi  26560  uhgr2edg  29240  umgrvad2edg  29245  uspgr2v1e2w  29283  usgr2v1e2w  29284  nb3grprlem1  29412  nb3grprlem2  29413  1hegrvtxdg1  29540  cyc3genpmlem  33154  elrspunsn  33437  prsiga  34112  bj-prmoore  37098  ftc1anclem8  37687  pr2el2  43541  pr2eldif2  43545  fourierdlem54  46116  prsal  46274  sge0pr  46350  imarnf1pr  47232  paireqne  47436  stgrnbgr0  47867  1hegrlfgr  47976  lubprlem  48759