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Theorem tpid1g 3582
Description: Closed theorem form of tpid1 3581. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
tpid1g (𝐴𝐵𝐴 ∈ {𝐴, 𝐶, 𝐷})

Proof of Theorem tpid1g
StepHypRef Expression
1 eqid 2100 . . 3 𝐴 = 𝐴
213mix1i 1121 . 2 (𝐴 = 𝐴𝐴 = 𝐶𝐴 = 𝐷)
3 eltpg 3516 . 2 (𝐴𝐵 → (𝐴 ∈ {𝐴, 𝐶, 𝐷} ↔ (𝐴 = 𝐴𝐴 = 𝐶𝐴 = 𝐷)))
42, 3mpbiri 167 1 (𝐴𝐵𝐴 ∈ {𝐴, 𝐶, 𝐷})
Colors of variables: wff set class
Syntax hints:  wi 4  w3o 929   = wceq 1299  wcel 1448  {ctp 3476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3or 931  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-tp 3482
This theorem is referenced by:  rngbaseg  11857  srngbased  11864  lmodbased  11875  ipsbased  11883  ipsscad  11886  topgrpbasd  11893
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