Theorem tpid3g 4731

Description: Closed theorem form of tpid3 4732. (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 1349	. 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴)
3		eltpg 4645	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐴} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴)))
4	2, 3	mpbiri 260	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

Syntax hints:   → wi 4   ∨ w3o 1097   = wceq 1560   ∈ wcel 2142  {ctp 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-sn 4583  df-pr 4585  df-tp 4587

This theorem is referenced by:  tpid3  4732  f1dom3fv3dif  7252  f1dom3el3dif  7253  en3lplem1  9567  en3lp  9569  tpf  14512  nb3grprlem1  29581  cplgr3v  29636  cshw1s2  33138  cyc3co2  33320  en3lplem1VD  45418  en3lpVD  45420  limsupequzlem  46296  etransclem48  46856