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  28465  umgrvad2edg  28470  uspgr2v1e2w  28508  usgr2v1e2w  28509  nbusgredgeu0  28625  nbusgrf1o0  28626  nb3grprlem1  28637  nb3grprlem2  28638  vtxduhgr0nedg  28749  1hegrvtxdg1  28764  1egrvtxdg1  28766  umgr2v2evd2  28784  vdegp1bi  28794  mptprop  31920  altgnsg  32308  cyc3genpmlem  32310  elrspunsn  32547  bj-prmoore  35996  ftc1anclem8  36568  kelac2  41807  pr2el1  42300  pr2eldif1  42305  fourierdlem54  44876  sge0pr  45110  imarnf1pr  45990  paireqne  46179  fmtnoprmfac2lem1  46234  1hegrlfgr  46510