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Theorem tpid3g 4439
 Description: Closed theorem form of tpid3 4440. (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 2770 . . 3 𝐴 = 𝐴
213mix3i 1418 . 2 (𝐴 = 𝐶𝐴 = 𝐷𝐴 = 𝐴)
3 eltpg 4362 . 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 1069   = wceq 1630   ∈ wcel 2144  {ctp 4318 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-un 3726  df-sn 4315  df-pr 4317  df-tp 4319 This theorem is referenced by:  tpid3  4440  tpnzd  4446  f1dom3fv3dif  6667  f1dom3el3dif  6668  en3lplem1  8670  en3lp  8672  nb3grprlem1  26504  cplgr3v  26565  en3lplem1VD  39594  en3lpVD  39596  limsupequzlem  40466  etransclem48  41010
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