Theorem prid2g 4786

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 4785	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵, 𝐴})
2		prcom 4757	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
3	1, 2	eleqtrdi 2854	1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∈ wcel 2108  {cpr 4650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651

This theorem is referenced by:  prel12g  4888  prproe  4929  unisn2  5330  fr2nr  5677  fpr2g  7248  f1prex  7320  fvf1pr  7343  pw2f1olem  9142  hashprdifel  14447  gcdcllem3  16547  mgm2nsgrplem1  18953  mgm2nsgrplem2  18954  mgm2nsgrplem3  18955  sgrp2nmndlem1  18958  sgrp2rid2  18961  pmtrprfv  19495  m2detleib  22658  indistopon  23029  pptbas  23036  coseq0negpitopi  26563  uhgr2edg  29243  umgrvad2edg  29248  uspgr2v1e2w  29286  usgr2v1e2w  29287  nb3grprlem1  29415  nb3grprlem2  29416  1hegrvtxdg1  29543  cyc3genpmlem  33144  elrspunsn  33422  prsiga  34095  bj-prmoore  37081  ftc1anclem8  37660  pr2el2  43513  pr2eldif2  43517  fourierdlem54  46081  prsal  46239  sge0pr  46315  imarnf1pr  47197  paireqne  47385  1hegrlfgr  47855  lubprlem  48642