Theorem tpid3gVD 43905

Description: Virtual deduction proof of tpid3g 4775. (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 43676	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
2		3mix3 1330	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴))
3	1, 2	e2 43694	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)   )
4		abid 2711	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)} ↔ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴))
5	3, 4	e2bir 43696	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)}   )
6		dftp2 4692	. . . . . . . . 9 ⊢ {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)}
7	6	eleq2i 2823	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)})
8	5, 7	e2bir 43696	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝐶, 𝐷, 𝐴}   )
9		eleq1 2819	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
10	9	biimpd 228	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
11	1, 8, 10	e22 43734	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
12	11	in2 43668	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
13	12	gen11 43679	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
14		19.23v 1943	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
15	13, 14	e1bi 43692	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})   )
16		idn1 43637	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
17		elisset 2813	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
18	16, 17	e1a 43690	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ∃𝑥 𝑥 = 𝐴   )
19		id 22	. . 3 ⊢ ((∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) → (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
20	15, 18, 19	e11 43751	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   )
21	20	in1 43634	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

Syntax hints:   → wi 4   ∨ w3o 1084  ∀wal 1537   = wceq 1539  ∃wex 1779   ∈ wcel 2104  {cab 2707  {ctp 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-un 3952  df-sn 4628  df-pr 4630  df-tp 4632  df-vd1 43633  df-vd2 43641

This theorem is referenced by: (None)