Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  prid1g Structured version   Visualization version   GIF version

Theorem prid1g 4688
 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 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})

Proof of Theorem prid1g
StepHypRef Expression
1 eqid 2819 . . 3 𝐴 = 𝐴
21orci 861 . 2 (𝐴 = 𝐴𝐴 = 𝐵)
3 elprg 4580 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐴, 𝐵} ↔ (𝐴 = 𝐴𝐴 = 𝐵)))
42, 3mpbiri 260 1 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 843   = wceq 1530   ∈ wcel 2107  {cpr 4561 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-un 3939  df-sn 4560  df-pr 4562 This theorem is referenced by:  prid2g  4689  prid1  4690  prnzg  4705  preq1b  4769  prel12g  4786  elpreqprb  4790  prproe  4828  opth1  5358  fr2nr  5526  fpr2g  6966  f1prex  7032  fveqf1o  7050  pw2f1olem  8613  hashprdifel  13751  gcdcllem3  15842  mgm2nsgrplem1  18075  mgm2nsgrplem2  18076  mgm2nsgrplem3  18077  sgrp2nmndlem1  18080  sgrp2rid2  18083  pmtrprfv  18573  pptbas  21608  coseq0negpitopi  25081  uhgr2edg  26982  umgrvad2edg  26987  uspgr2v1e2w  27025  usgr2v1e2w  27026  nbusgredgeu0  27142  nbusgrf1o0  27143  nb3grprlem1  27154  nb3grprlem2  27155  vtxduhgr0nedg  27266  1hegrvtxdg1  27281  1egrvtxdg1  27283  umgr2v2evd2  27301  vdegp1bi  27311  mptprop  30426  altgnsg  30784  cyc3genpmlem  30786  bj-prmoore  34394  ftc1anclem8  34961  kelac2  39650  pr2el1  39893  pr2eldif1  39898  fourierdlem54  42430  sge0pr  42661  imarnf1pr  43466  paireqne  43658  fmtnoprmfac2lem1  43713  1hegrlfgr  43992
 Copyright terms: Public domain W3C validator