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Theorem tpid3g 4248
 Description: Closed theorem form of tpid3 4250. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 30-Apr-2021.)
Assertion
Ref Expression
tpid3g (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3g
StepHypRef Expression
1 eqid 2610 . . 3 𝐴 = 𝐴
213mix3i 1228 . 2 (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐴)
3 eltpg 4174 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐴} ↔ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐴)))
42, 3mpbiri 247 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977  {ctp 4129 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128  df-tp 4130 This theorem is referenced by:  tpid3  4250  tpnzd  4257  f1dom3fv3dif  6425  f1dom3el3dif  6426  en3lplem1  8394  en3lp  8396  nb3graprlem1  25980  en3lplem1VD  38100  en3lpVD  38102  etransclem48  39175  nb3grprlem1  40608  cplgr3v  40657
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