Theorem tpid3g 4737

Description: Closed theorem form of tpid3 4738. (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 2762	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix3i 1353	. 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴)
3		eltpg 4651	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐴} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴)))
4	2, 3	mpbiri 261	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

This proof depends on syntax axioms:   → wi 4   ∨ w3o 1101   = wceq 1569   ∈ wcel 2142  {ctp 4592

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-sn 4589  df-pr 4591  df-tp 4593

This theorem is used by:  tpid3  4738  f1dom3fv3dif  7266  f1dom3el3dif  7267  en3lplem1  9579  en3lp  9581  tpf  14543  nb3grprlem1  29741  cplgr3v  29796  cshw1s2  33289  cyc3co2  33469  en3lplem1VD  45579  en3lpVD  45581  limsupequzlem  46464  etransclem48  47024