Theorem tpid2g 4735

Description: Closed theorem form of tpid2 4734. (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 2762	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix2i 1353	. 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷)
3		eltpg 4650	. 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})

This proof depends on syntax axioms:   → wi 4   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145  {ctp 4591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-sn 4588  df-pr 4590  df-tp 4592

This theorem is used by:  tpf  14564  cplgr3v  29879  cshw1s2  33385  cyc3co2  33565  limsupequzlem  46535