Theorem tpid2g 4743

Description: Closed theorem form of tpid2 4742. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Assertion

Ref	Expression
tpid2g	⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷})

Proof of Theorem tpid2g

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix2i 1335	. 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷)
3		eltpg 4658	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷)))
4	2, 3	mpbiri 258	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷})

Syntax hints:   → wi 4   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  {ctp 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3457  df-un 3927  df-sn 4598  df-pr 4600  df-tp 4602

This theorem is referenced by:  tpf  14474  cplgr3v  29369  cshw1s2  32890  cyc3co2  33105  limsupequzlem  45693