Theorem tpid2g 39316

Description: Closed theorem form of tpid2 4304. (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 2622	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix2i 1234	. 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷)
3		eltpg 4227	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷)))
4	2, 3	mpbiri 248	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷})

Syntax hints:   → wi 4   ∨ w3o 1036   = wceq 1483   ∈ wcel 1990  {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  limsupequzlem  39954