Theorem prid1g 4765

Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Assertion

Ref	Expression
prid1g	⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})

Proof of Theorem prid1g

Step	Hyp	Ref	Expression
1		eqid 2733	. . 3 ⊢ 𝐴 = 𝐴
2	1	orci 864	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
3		elprg 4650	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴, 𝐵} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)))
4	2, 3	mpbiri 258	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∨ wo 846   = wceq 1542   ∈ 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:  prid2g  4766  prid1  4767  prnzg  4783  preq1b  4848  prel12g  4865  elpreqprb  4869  prproe  4907  opth1  5476  fr2nr  5655  fpr2g  7213  f1prex  7282  fveqf1o  7301  pw2f1olem  9076  hashprdifel  14358  gcdcllem3  16442  mgm2nsgrplem1  18799  mgm2nsgrplem2  18800  mgm2nsgrplem3  18801  sgrp2nmndlem1  18804  sgrp2rid2  18807  pmtrprfv  19321  pptbas  22511  coseq0negpitopi  26013  uhgr2edg  28496  umgrvad2edg  28501  uspgr2v1e2w  28539  usgr2v1e2w  28540  nbusgredgeu0  28656  nbusgrf1o0  28657  nb3grprlem1  28668  nb3grprlem2  28669  vtxduhgr0nedg  28780  1hegrvtxdg1  28795  1egrvtxdg1  28797  umgr2v2evd2  28815  vdegp1bi  28825  mptprop  31951  altgnsg  32339  cyc3genpmlem  32341  elrspunsn  32578  bj-prmoore  36044  ftc1anclem8  36616  kelac2  41855  pr2el1  42348  pr2eldif1  42353  fourierdlem54  44924  sge0pr  45158  imarnf1pr  46038  paireqne  46227  fmtnoprmfac2lem1  46282  1hegrlfgr  46558