Theorem prid1g 4785

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 2740	. . 3 ⊢ 𝐴 = 𝐴
2	1	orci 864	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
3		elprg 4670	. 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 1537   ∈ wcel 2108  {cpr 4650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651

This theorem is referenced by:  prid2g  4786  prid1  4787  prnzg  4803  preq1b  4871  prel12g  4888  elpreqprb  4892  prproe  4929  opth1  5495  fr2nr  5677  fpr2g  7248  f1prex  7320  fveqf1o  7338  fvf1pr  7343  pw2f1olem  9142  hashprdifel  14447  gcdcllem3  16547  mgm2nsgrplem1  18953  mgm2nsgrplem2  18954  mgm2nsgrplem3  18955  sgrp2nmndlem1  18958  sgrp2rid2  18961  pmtrprfv  19495  pptbas  23036  coseq0negpitopi  26563  uhgr2edg  29243  umgrvad2edg  29248  uspgr2v1e2w  29286  usgr2v1e2w  29287  nbusgredgeu0  29403  nbusgrf1o0  29404  nb3grprlem1  29415  nb3grprlem2  29416  vtxduhgr0nedg  29528  1hegrvtxdg1  29543  1egrvtxdg1  29545  umgr2v2evd2  29563  vdegp1bi  29573  mptprop  32710  altgnsg  33142  cyc3genpmlem  33144  elrspunsn  33422  bj-prmoore  37081  ftc1anclem8  37660  kelac2  43022  pr2el1  43511  pr2eldif1  43516  fourierdlem54  46081  sge0pr  46315  imarnf1pr  47197  paireqne  47385  fmtnoprmfac2lem1  47440  1hegrlfgr  47855