Theorem en3lplem1VD 43906

Description: Virtual deduction proof of en3lplem1 9609. (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 43637	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   )
2		simp3 1136	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴)
3	1, 2	e1a 43690	. . . . . 6 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   𝐶 ∈ 𝐴   )
4		tpid3g 4775	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
5	3, 4	e1a 43690	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   𝐶 ∈ {𝐴, 𝐵, 𝐶}   )
6		idn2 43676	. . . . . 6 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7		eleq2 2820	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐶 ∈ 𝑥 ↔ 𝐶 ∈ 𝐴))
8	7	biimprd 247	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝑥))
9	6, 3, 8	e21 43793	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 = 𝐴   ▶   𝐶 ∈ 𝑥   )
10		pm3.2 468	. . . . 5 ⊢ (𝐶 ∈ {𝐴, 𝐵, 𝐶} → (𝐶 ∈ 𝑥 → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ 𝑥)))
11	5, 9, 10	e12 43787	. . . 4 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 = 𝐴   ▶   (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ 𝑥)   )
12		elex22 3495	. . . 4 ⊢ ((𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ 𝑥) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))
13	11, 12	e2 43694	. . 3 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 = 𝐴   ▶   ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)   )
14	13	in2 43668	. 2 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))   )
15	14	in1 43634	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539  ∃wex 1779   ∈ wcel 2104  {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-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  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:  en3lplem2VD  43907