Theorem tpid1g 3630

Description: Closed theorem form of tpid1 3629. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Assertion

Ref	Expression
tpid1g	⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷})

Proof of Theorem tpid1g

Step	Hyp	Ref	Expression
1		eqid 2137	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix1i 1153	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)
3		eltpg 3564	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐴, 𝐶, 𝐷} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
4	2, 3	mpbiri 167	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷})

Syntax hints:   → wi 4   ∨ w3o 961   = wceq 1331   ∈ wcel 1480  {ctp 3524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3or 963  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-tp 3530

This theorem is referenced by:  rngbaseg  12064  srngbased  12071  lmodbased  12082  ipsbased  12090  ipsscad  12093  topgrpbasd  12100