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 2735	. . 3 ⊢ 𝐴 = 𝐴
2	1	orci 865	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
3		elprg 4653	. 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 847   = wceq 1537   ∈ 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:  prid2g  4766  prid1  4767  prnzg  4783  preq1b  4851  prel12g  4869  elpreqprb  4873  prproe  4910  opth1  5486  fr2nr  5666  fpr2g  7231  f1prex  7304  fveqf1o  7322  fvf1pr  7327  pw2f1olem  9115  hashprdifel  14434  gcdcllem3  16535  mgm2nsgrplem1  18944  mgm2nsgrplem2  18945  mgm2nsgrplem3  18946  sgrp2nmndlem1  18949  sgrp2rid2  18952  pmtrprfv  19486  pptbas  23031  coseq0negpitopi  26560  uhgr2edg  29240  umgrvad2edg  29245  uspgr2v1e2w  29283  usgr2v1e2w  29284  nbusgredgeu0  29400  nbusgrf1o0  29401  nb3grprlem1  29412  nb3grprlem2  29413  vtxduhgr0nedg  29525  1hegrvtxdg1  29540  1egrvtxdg1  29542  umgr2v2evd2  29560  vdegp1bi  29570  mptprop  32713  altgnsg  33152  cyc3genpmlem  33154  elrspunsn  33437  bj-prmoore  37098  ftc1anclem8  37687  kelac2  43054  pr2el1  43539  pr2eldif1  43544  fourierdlem54  46116  sge0pr  46350  imarnf1pr  47232  paireqne  47436  fmtnoprmfac2lem1  47491  1hegrlfgr  47976