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Theorem tpid3g 4296
 Description: Closed theorem form of tpid3 4298. (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 2620 . . 3 𝐴 = 𝐴
213mix3i 1233 . 2 (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐴)
3 eltpg 4218 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐴} ↔ (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐴)))
42, 3mpbiri 248 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1035   = wceq 1481   ∈ wcel 1988  {ctp 4172 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-un 3572  df-sn 4169  df-pr 4171  df-tp 4173 This theorem is referenced by:  tpid3  4298  tpnzd  4305  f1dom3fv3dif  6510  f1dom3el3dif  6511  en3lplem1  8496  en3lp  8498  nb3grprlem1  26263  cplgr3v  26312  en3lplem1VD  38898  en3lpVD  38900  limsupequzlem  39754  etransclem48  40262
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