Theorem tpid3gVD 46013

Description: Virtual deduction proof of tpid3g 4733. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tpid3gVD	⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3gVD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn2 45785	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
2		3mix3 1351	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴))
3	1, 2	e2 45803	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)   )
4		abid 2745	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)} ↔ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴))
5	3, 4	e2bir 45805	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)}   )
6		dftp2 4652	. . . . . . . . 9 ⊢ {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)}
7	6	eleq2i 2856	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)})
8	5, 7	e2bir 45805	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝐶, 𝐷, 𝐴}   )
9		eleq1 2852	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
10	9	biimpd 232	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
11	1, 8, 10	e22 45843	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
12	11	in2 45777	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
13	12	gen11 45788	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
14		19.23v 1977	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
15	13, 14	e1bi 45801	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
16		idn1 45746	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
17		elisset 2846	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
18	16, 17	e1a 45799	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ∃𝑥 𝑥 = 𝐴   )
19		id 23	. . 3 ⊢ ((∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) → (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
20	15, 18, 19	e11 45860	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
21	20	in1 45743	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

This proof depends on syntax axioms:   → wi 4   ∨ w3o 1102  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  {ctp 4592

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-un 4154  df-sn 4589  df-pr 4591  df-tp 4593  df-vd1 45742  df-vd2 45750

This theorem is used by: (None)