Theorem tpid1g 4705

Description: Closed theorem form of tpid1 4704. (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 2738	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix1i 1332	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)
3		eltpg 4621	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐴, 𝐶, 𝐷} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
4	2, 3	mpbiri 257	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷})

Syntax hints:   → wi 4   ∨ w3o 1085   = wceq 1539   ∈ wcel 2106  {ctp 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-sn 4562  df-pr 4564  df-tp 4566

This theorem is referenced by:  tpnzd  4716  cplgr3v  27802  cyc3co2  31407  limsupequzlem  43263