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Theorem tpid1g 4665
 Description: Closed theorem form of tpid1 4664. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
tpid1g (𝐴𝐵𝐴 ∈ {𝐴, 𝐶, 𝐷})

Proof of Theorem tpid1g
StepHypRef Expression
1 eqid 2758 . . 3 𝐴 = 𝐴
213mix1i 1330 . 2 (𝐴 = 𝐴𝐴 = 𝐶𝐴 = 𝐷)
3 eltpg 4583 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐴, 𝐶, 𝐷} ↔ (𝐴 = 𝐴𝐴 = 𝐶𝐴 = 𝐷)))
42, 3mpbiri 261 1 (𝐴𝐵𝐴 ∈ {𝐴, 𝐶, 𝐷})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111  {ctp 4529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3865  df-sn 4526  df-pr 4528  df-tp 4530 This theorem is referenced by:  tpnzd  4676  cplgr3v  27329  cyc3co2  30937  limsupequzlem  42758
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