Theorem tpid3g 4739

Description: Closed theorem form of tpid3 4740. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 30-Apr-2021.)

Assertion

Ref	Expression
tpid3g	⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3g

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix3i 1336	. 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴)
3		eltpg 4653	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐴} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴)))
4	2, 3	mpbiri 258	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

Syntax hints:   → wi 4   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  {ctp 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-un 3922  df-sn 4593  df-pr 4595  df-tp 4597

This theorem is referenced by:  tpid3  4740  f1dom3fv3dif  7246  f1dom3el3dif  7247  en3lplem1  9572  en3lp  9574  tpf  14471  nb3grprlem1  29314  cplgr3v  29369  cshw1s2  32889  cyc3co2  33104  en3lplem1VD  44839  en3lpVD  44841  limsupequzlem  45727  etransclem48  46287