Theorem tpid3g 4729

Description: Closed theorem form of tpid3 4730. (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 2760	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix3i 1354	. 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴)
3		eltpg 4643	. 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 1102   = wceq 1569   ∈ wcel 2144  {ctp 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 4150  df-sn 4585  df-pr 4587  df-tp 4589

This theorem is used by:  tpid3  4730  f1dom3fv3dif  7276  f1dom3el3dif  7277  en3lplem1  9620  en3lp  9622  tpf  14651  nb3grprlem1  29970  cplgr3v  30025  cshw1s2  33571  cyc3co2  33751  en3lplem1VD  45961  en3lpVD  45963  limsupequzlem  46868  etransclem48  47428