Theorem tpid1g 4727

Description: Closed theorem form of tpid1 4726. (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 2761	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix1i 1346	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)
3		eltpg 4644	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐴, 𝐶, 𝐷} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
4	2, 3	mpbiri 260	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷})

Syntax hints:   → wi 4   ∨ w3o 1096   = wceq 1559   ∈ wcel 2141  {ctp 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-sn 4582  df-pr 4584  df-tp 4586

This theorem is referenced by:  tpnzd  4738  tpf  14509  cplgr3v  29582  cyc3co2  33281  limsupequzlem  46260