Theorem en3lplem1VD 45025

Description: Virtual deduction proof of en3lplem1 9519. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
en3lplem1VD	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem en3lplem1VD

Step	Hyp	Ref	Expression
1		idn1 44757	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   )
2		simp3 1138	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴)
3	1, 2	e1a 44810	. . . . . 6 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   𝐶 ∈ 𝐴   )
4		tpid3g 4727	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
5	3, 4	e1a 44810	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   𝐶 ∈ {𝐴, 𝐵, 𝐶}   )
6		idn2 44796	. . . . . 6 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7		eleq2 2823	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐶 ∈ 𝑥 ↔ 𝐶 ∈ 𝐴))
8	7	biimprd 248	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝑥))
9	6, 3, 8	e21 44912	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 = 𝐴   ▶   𝐶 ∈ 𝑥   )
10		pm3.2 469	. . . . 5 ⊢ (𝐶 ∈ {𝐴, 𝐵, 𝐶} → (𝐶 ∈ 𝑥 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ 𝑥)))
11	5, 9, 10	e12 44906	. . . 4 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 = 𝐴   ▶   (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ 𝑥)   )
12		elex22 3463	. . . 4 ⊢ ((𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ 𝑥) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))
13	11, 12	e2 44814	. . 3 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 = 𝐴   ▶   ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)   )
14	13	in2 44788	. 2 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))   )
15	14	in1 44754	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {ctp 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-un 3904  df-sn 4579  df-pr 4581  df-tp 4583  df-vd1 44753  df-vd2 44761

This theorem is referenced by:  en3lplem2VD  45026