Theorem prid1g 3776

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 2230	. . 3 ⊢ 𝐴 = 𝐴
2	1	orci 738	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵)
3		elprg 3690	. 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 2201  {cpr 3671

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 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-un 3203  df-sn 3676  df-pr 3677

This theorem is referenced by:  prid2g  3777  prid1  3778  preqr1g  3850  opth1  4330  en2lp  4654  acexmidlemcase  6018  pw2f1odclem  7025  en2eqpr  7104  m1expcl2  10829  maxabslemval  11791  xrmaxiflemval  11833  xrmaxaddlem  11843  2strbasg  13226  2strbas1g  13229  coseq0negpitopi  15589  structvtxval  15919  umgrnloopv  15994  umgredgprv  15995  umgrpredgv  16027  uhgr2edg  16086  umgrvad2edg  16091  usgr2v1e2w  16126  1hegrvtxdg1fi  16189  vdegp1bid  16195