Theorem eltpi 4649

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 4647	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
2	1	ibi 270	1 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))

This proof depends on syntax axioms:   → wi 4   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145  {ctp 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-sn 4585  df-pr 4587  df-tp 4589

This theorem is used by:  fvf1tp  13909  tpfo  14625  sgnmulsgn  15242  prm23lt5  16972  perfectlem2  27539  zabsle1  27605  sgnmulsgp  33405  gsumtp  33607  cyc3co2  33683  kur14lem7  35946  omcl3g  44294  fmtnofz04prm  48606  perfectALTVlem2  48764  gpgprismgr4cycllem7  49143  pgnbgreunbgrlem3  49160  pgnbgreunbgrlem6  49166  crosspaltd  50910  crossp3d  50911