Theorem en3lplem2VD 45203

Description: Virtual deduction proof of en3lplem2 9534. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
en3lplem2VD	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)))

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

Proof of Theorem en3lplem2VD

Step	Hyp	Ref	Expression
1		idn1 44934	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   )
2		idn3 44975	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
3		en3lplem1VD 45202	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)))
4	1, 2, 3	e13 45107	. . . . . 6 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐴   ▶   ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)   )
5	4	in3 44969	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))   )
6		3anrot 1100	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) ↔ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵))
7	1, 6	e1bi 44989	. . . . . . . 8 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)   )
8		idn3 44975	. . . . . . . 8 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   𝑥 = 𝐵   )
9		en3lplem1VD 45202	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝑥 = 𝐵 → ∃𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦 ∈ 𝑥)))
10	7, 8, 9	e13 45107	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   ∃𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦 ∈ 𝑥)   )
11		tprot 4708	. . . . . . . . . 10 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
12	11	eleq2i 2829	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑦 ∈ {𝐵, 𝐶, 𝐴})
13	12	anbi1i 625	. . . . . . . 8 ⊢ ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦 ∈ 𝑥))
14	13	exbii 1850	. . . . . . 7 ⊢ (∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥) ↔ ∃𝑦(𝑦 ∈ {𝐵, 𝐶, 𝐴} ∧ 𝑦 ∈ 𝑥))
15	10, 14	e3bir 45098	. . . . . 6 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐵   ▶   ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)   )
16	15	in3 44969	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐵 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))   )
17		jao 963	. . . . 5 ⊢ ((𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)) → ((𝑥 = 𝐵 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))))
18	5, 16, 17	e22 45031	. . . 4 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))   )
19		3anrot 1100	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
20	1, 19	e1bir 44990	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   (𝐶 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶)   )
21		idn3 44975	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   𝑥 = 𝐶   )
22		en3lplem1VD 45202	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → (𝑥 = 𝐶 → ∃𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦 ∈ 𝑥)))
23	20, 21, 22	e13 45107	. . . . . 6 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   ∃𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦 ∈ 𝑥)   )
24		tprot 4708	. . . . . . . . 9 ⊢ {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
25	24	eleq2i 2829	. . . . . . . 8 ⊢ (𝑦 ∈ {𝐶, 𝐴, 𝐵} ↔ 𝑦 ∈ {𝐴, 𝐵, 𝐶})
26	25	anbi1i 625	. . . . . . 7 ⊢ ((𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))
27	26	exbii 1850	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ {𝐶, 𝐴, 𝐵} ∧ 𝑦 ∈ 𝑥) ↔ ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))
28	23, 27	e3bi 45097	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ,   𝑥 = 𝐶   ▶   ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)   )
29	28	in3 44969	. . . 4 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐶 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))   )
30		idn2 44973	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   𝑥 ∈ {𝐴, 𝐵, 𝐶}   )
31		dftp2 4650	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}
32	31	eleq2i 2829	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)})
33	30, 32	e2bi 44992	. . . . . 6 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}   )
34		abid 2719	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶))
35	33, 34	e2bi 44992	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)   )
36		df-3or 1088	. . . . 5 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) ↔ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
37	35, 36	e2bi 44992	. . . 4 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∨ 𝑥 = 𝐶)   )
38		jao 963	. . . 4 ⊢ (((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)) → ((𝑥 = 𝐶 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)) → (((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∨ 𝑥 = 𝐶) → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))))
39	18, 29, 37, 38	e222 44996	. . 3 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ,   𝑥 ∈ {𝐴, 𝐵, 𝐶}   ▶   ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)   )
40	39	in2 44965	. 2 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   ▶   (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))   )
41	40	in1 44931	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∨ w3o 1086   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  {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-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  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  df-vd1 44930  df-vd2 44938  df-vd3 44950

This theorem is referenced by:  en3lpVD  45204