Theorem eltpi 4692

Description: A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
eltpi	⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))

Proof of Theorem eltpi

Step	Hyp	Ref	Expression
1		eltpg 4690	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
2	1	ibi 267	1 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))

Syntax hints:   → wi 4   ∨ w3o 1084   = wceq 1534   ∈ wcel 2099  {ctp 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-un 3952  df-sn 4630  df-pr 4632  df-tp 4634

This theorem is referenced by:  prm23lt5  16782  perfectlem2  27162  zabsle1  27228  cyc3co2  32861  sgnmulsgn  34169  sgnmulsgp  34170  kur14lem7  34822  omcl3g  42763  fmtnofz04prm  46917  perfectALTVlem2  47062