Theorem en3lplem2 9076

Description: Lemma for en3lp 9077. (Contributed by Alan Sare, 28-Oct-2011.)

Assertion

Ref	Expression
en3lplem2	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

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

Proof of Theorem en3lplem2

Step	Hyp	Ref	Expression
1		en3lplem1 9075	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
2		en3lplem1 9075	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅))
3	2	3comr 1121	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅))
4	3	a1d 25	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅)))
5		tprot 4685	. . . . . . . . 9 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
6	5	ineq2i 4186	. . . . . . . 8 ⊢ (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = (𝑥 ∩ {𝐵, 𝐶, 𝐴})
7	6	neeq1i 3080	. . . . . . 7 ⊢ ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅ ↔ (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅)
8	7	bicomi 226	. . . . . 6 ⊢ ((𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅ ↔ (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
9	4, 8	syl8ib 258	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐵 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
10		jao 957	. . . . 5 ⊢ ((𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅) → ((𝑥 = 𝐵 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
11	1, 9, 10	sylsyld 61	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
12	11	imp 409	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
13		en3lplem1 9075	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅))
14	13	3coml 1123	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅))
15	14	a1d 25	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅)))
16		tprot 4685	. . . . . . 7 ⊢ {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
17	16	ineq2i 4186	. . . . . 6 ⊢ (𝑥 ∩ {𝐶, 𝐴, 𝐵}) = (𝑥 ∩ {𝐴, 𝐵, 𝐶})
18	17	neeq1i 3080	. . . . 5 ⊢ ((𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅ ↔ (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
19	15, 18	syl8ib 258	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐶 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
20	19	imp 409	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 = 𝐶 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
21		idd 24	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → 𝑥 ∈ {𝐴, 𝐵, 𝐶}))
22		dftp2 4627	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}
23	22	eleq2i 2904	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)})
24	21, 23	syl6ib 253	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}))
25		abid 2803	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶))
26	24, 25	syl6ib 253	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)))
27		df-3or 1084	. . . . 5 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) ↔ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
28	26, 27	syl6ib 253	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∨ 𝑥 = 𝐶)))
29	28	imp 409	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
30	12, 20, 29	mpjaod 856	. 2 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
31	30	ex 415	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   ∨ w3o 1082   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {cab 2799   ≠ wne 3016   ∩ cin 3935  ∅c0 4291  {ctp 4571

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-nul 4292  df-sn 4568  df-pr 4570  df-tp 4572

This theorem is referenced by:  en3lp  9077