Theorem prid2g 4742

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 4741	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵, 𝐴})
2		prcom 4713	. 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵}
3	1, 2	eleqtrdi 2845	1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∈ wcel 2109  {cpr 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-sn 4607  df-pr 4609

This theorem is referenced by:  prel12g  4845  prproe  4886  unisn2  5287  fr2nr  5636  fpr2g  7208  f1prex  7282  fvf1pr  7305  pw2f1olem  9095  hashprdifel  14421  gcdcllem3  16525  mgm2nsgrplem1  18901  mgm2nsgrplem2  18902  mgm2nsgrplem3  18903  sgrp2nmndlem1  18906  sgrp2rid2  18909  pmtrprfv  19439  m2detleib  22574  indistopon  22944  pptbas  22951  coseq0negpitopi  26469  uhgr2edg  29192  umgrvad2edg  29197  uspgr2v1e2w  29235  usgr2v1e2w  29236  nb3grprlem1  29364  nb3grprlem2  29365  1hegrvtxdg1  29492  cyc3genpmlem  33167  elrspunsn  33449  prsiga  34167  bj-prmoore  37138  ftc1anclem8  37729  pr2el2  43550  pr2eldif2  43554  fourierdlem54  46169  prsal  46327  sge0pr  46403  imarnf1pr  47291  paireqne  47505  stgrnbgr0  47956  1hegrlfgr  48087  lubprlem  48916  2arwcatlem4  49455  2arwcat  49457  incat  49458