Theorem eltpi 4641

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 4639	. 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 1085   = wceq 1541   ∈ wcel 2111  {ctp 4580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3907  df-sn 4577  df-pr 4579  df-tp 4581

This theorem is referenced by:  fvf1tp  13690  tpfo  14404  prm23lt5  16723  perfectlem2  27166  zabsle1  27232  sgnmulsgn  32820  sgnmulsgp  32821  gsumtp  33033  cyc3co2  33104  kur14lem7  35244  omcl3g  43366  fmtnofz04prm  47607  perfectALTVlem2  47752  gpgprismgr4cycllem7  48131  pgnbgreunbgrlem3  48148  pgnbgreunbgrlem6  48154