Theorem prid1g 3775

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 2231	. . 3 ⊢ 𝐴 = 𝐴
2	1	orci 738	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
3		elprg 3689	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴, 𝐵} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)))
4	2, 3	mpbiri 168	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})

Syntax hints:   → wi 4   ∨ wo 715   = wceq 1397   ∈ wcel 2202  {cpr 3670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  prid2g  3776  prid1  3777  preqr1g  3849  opth1  4328  en2lp  4652  acexmidlemcase  6013  pw2f1odclem  7020  en2eqpr  7099  m1expcl2  10824  maxabslemval  11770  xrmaxiflemval  11812  xrmaxaddlem  11822  2strbasg  13205  2strbas1g  13208  coseq0negpitopi  15563  structvtxval  15893  umgrnloopv  15968  umgredgprv  15969  umgrpredgv  16001  uhgr2edg  16060  umgrvad2edg  16065  usgr2v1e2w  16100  1hegrvtxdg1fi  16163  vdegp1bid  16169