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Theorem eltpi 4684
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 4682 . 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 1083   = wceq 1533  wcel 2098  {ctp 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3946  df-sn 4622  df-pr 4624  df-tp 4626
This theorem is referenced by:  prm23lt5  16752  perfectlem2  27103  zabsle1  27169  cyc3co2  32792  sgnmulsgn  34067  sgnmulsgp  34068  kur14lem7  34720  omcl3g  42633  fmtnofz04prm  46790  perfectALTVlem2  46935
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