Theorem tpid1g 3746

Description: Closed theorem form of tpid1 3745. (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 2206	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix1i 1172	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)
3		eltpg 3679	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐴, 𝐶, 𝐷} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
4	2, 3	mpbiri 168	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷})

Syntax hints:   → wi 4   ∨ w3o 980   = wceq 1373   ∈ wcel 2177  {ctp 3636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3or 982  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3171  df-sn 3640  df-pr 3641  df-tp 3642

This theorem is referenced by:  rngbaseg  13012  srngbased  13023  lmodbased  13041  ipsbased  13053  ipsscad  13056  topgrpbasd  13073  psrbasg  14480