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 2737	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix3i 1337	. 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 258	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

Syntax hints:   → wi 4   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114  {ctp 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-sn 4583  df-pr 4585  df-tp 4587

This theorem is referenced by:  tpid3  4732  f1dom3fv3dif  7224  f1dom3el3dif  7225  en3lplem1  9533  en3lp  9535  tpf  14434  nb3grprlem1  29465  cplgr3v  29520  cshw1s2  33053  cyc3co2  33234  en3lplem1VD  45198  en3lpVD  45200  limsupequzlem  46080  etransclem48  46640