Theorem prid1g 4741

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 2736	. . 3 ⊢ 𝐴 = 𝐴
2	1	orci 865	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
3		elprg 4629	. 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 1540   ∈ 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:  prid2g  4742  prid1  4743  prnzg  4759  preq1b  4827  prel12g  4845  elpreqprb  4849  prproe  4886  opth1  5455  fr2nr  5636  fpr2g  7208  f1prex  7282  fveqf1o  7300  fvf1pr  7305  pw2f1olem  9095  hashprdifel  14421  gcdcllem3  16525  mgm2nsgrplem1  18901  mgm2nsgrplem2  18902  mgm2nsgrplem3  18903  sgrp2nmndlem1  18906  sgrp2rid2  18909  pmtrprfv  19439  pptbas  22951  coseq0negpitopi  26469  uhgr2edg  29192  umgrvad2edg  29197  uspgr2v1e2w  29235  usgr2v1e2w  29236  nbusgredgeu0  29352  nbusgrf1o0  29353  nb3grprlem1  29364  nb3grprlem2  29365  vtxduhgr0nedg  29477  1hegrvtxdg1  29492  1egrvtxdg1  29494  umgr2v2evd2  29512  vdegp1bi  29522  mptprop  32680  altgnsg  33165  cyc3genpmlem  33167  elrspunsn  33449  bj-prmoore  37138  ftc1anclem8  37729  kelac2  43064  pr2el1  43548  pr2eldif1  43553  fourierdlem54  46169  sge0pr  46403  imarnf1pr  47291  paireqne  47505  fmtnoprmfac2lem1  47560  1hegrlfgr  48087  termc2  49383  2arwcatlem4  49455  incat  49458