Theorem tpid2g 4667

Description: Closed theorem form of tpid2 4666. (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 2798	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix2i 1331	. 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷)
3		eltpg 4583	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷)))
4	2, 3	mpbiri 261	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷})

Syntax hints:   → wi 4   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111  {ctp 4529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-tp 4530

This theorem is referenced by:  cplgr3v  27225  cshw1s2  30660  cyc3co2  30832  limsupequzlem  42364