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Theorem tpid2g 4523
 Description: Closed theorem form of tpid2 4522. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
tpid2g (𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})

Proof of Theorem tpid2g
StepHypRef Expression
1 eqid 2824 . . 3 𝐴 = 𝐴
213mix2i 1439 . 2 (𝐴 = 𝐶𝐴 = 𝐴𝐴 = 𝐷)
3 eltpg 4445 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶𝐴 = 𝐴𝐴 = 𝐷)))
42, 3mpbiri 250 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1112   = wceq 1658   ∈ wcel 2166  {ctp 4400 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2802 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-v 3415  df-un 3802  df-sn 4397  df-pr 4399  df-tp 4401 This theorem is referenced by:  limsupequzlem  40748
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