Theorem tpid2g 4706

Description: Closed theorem form of tpid2 4705. (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 2741	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix2i 1342	. 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷)
3		eltpg 4621	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷)))
4	2, 3	mpbiri 260	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷})

Syntax hints:   → wi 4   ∨ w3o 1092   = wceq 1548   ∈ wcel 2121  {ctp 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-un 3890  df-sn 4559  df-pr 4561  df-tp 4563

This theorem is referenced by:  tpf  14456  cplgr3v  29526  cshw1s2  33043  cyc3co2  33225  limsupequzlem  46179