Theorem tpid3g 4738

Description: Closed theorem form of tpid3 4739. (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 2763	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix3i 1354	. 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴)
3		eltpg 4652	. 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})

Syntax hints:   → wi 4   ∨ w3o 1102   = wceq 1570   ∈ wcel 2143  {ctp 4593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3910  df-sn 4590  df-pr 4592  df-tp 4594

This theorem is referenced by:  tpid3  4739  f1dom3fv3dif  7266  f1dom3el3dif  7267  en3lplem1  9577  en3lp  9579  tpf  14532  nb3grprlem1  29730  cplgr3v  29785  cshw1s2  33280  cyc3co2  33460  en3lplem1VD  45571  en3lpVD  45573  limsupequzlem  46456  etransclem48  47016