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Theorem eltpi 4688
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 = 𝐵𝐴 = 𝐶𝐴 = 𝐷))

Proof of Theorem eltpi
StepHypRef Expression
1 eltpg 4686 . 2 (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷)))
21ibi 267 1 (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1086   = wceq 1540  wcel 2108  {ctp 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629  df-tp 4631
This theorem is referenced by:  fvf1tp  13829  tpfo  14539  prm23lt5  16852  perfectlem2  27274  zabsle1  27340  gsumtp  33061  cyc3co2  33160  sgnmulsgn  34552  sgnmulsgp  34553  kur14lem7  35217  omcl3g  43347  fmtnofz04prm  47564  perfectALTVlem2  47709
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