Theorem prid1g 4718

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 2761	. . 3 ⊢ 𝐴 = 𝐴
2	1	orci 876	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
3		elprg 4604	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴, 𝐵} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)))
4	2, 3	mpbiri 260	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∨ wo 858   = wceq 1559   ∈ wcel 2141  {cpr 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-sn 4582  df-pr 4584

This theorem is referenced by:  prid2g  4719  prid1  4720  prnzg  4736  preq1b  4803  prel12g  4821  elpreqprb  4825  prproe  4862  opth1  5442  fr2nr  5622  fpr2g  7191  f1prex  7264  fveqf1o  7282  fvf1pr  7287  pw2f1olem  9049  hashprdifel  14408  gcdcllem3  16518  mgm2nsgrplem1  18938  mgm2nsgrplem2  18939  mgm2nsgrplem3  18940  sgrp2nmndlem1  18943  sgrp2rid2  18946  pmtrprfv  19476  pptbas  23048  coseq0negpitopi  26545  uhgr2edg  29355  umgrvad2edg  29360  uspgr2v1e2w  29398  usgr2v1e2w  29399  nbusgredgeu0  29515  nbusgrf1o0  29516  nb3grprlem1  29527  nb3grprlem2  29528  vtxduhgr0nedg  29639  1hegrvtxdg1  29654  1egrvtxdg1  29656  umgr2v2evd2  29674  vdegp1bi  29684  mptprop  32850  altgnsg  33290  cyc3genpmlem  33292  elrspunsn  33576  esplyfval1  33831  bj-prmoore  37569  ftc1anclem8  38163  kelac2  43606  pr2el1  44089  pr2eldif1  44094  fourierdlem54  46698  sge0pr  46932  imarnf1pr  47840  paireqne  48081  fmtnoprmfac2lem1  48139  grlimprclnbgr  48582  grlimprclnbgredg  48583  1hegrlfgr  48718  fucoppcffth  49996  termc2  50103  uobeqterm  50131  2arwcatlem4  50183  incat  50186