Theorem en3lplem2 8512

Description: Lemma for en3lp 8513. (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 8511	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
2		en3lplem1 8511	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝑥 = 𝐵 → (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅))
3	2	3comr 1273	. . . . . . 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 4284	. . . . . . . . 9 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
6	5	ineq2i 3811	. . . . . . . 8 ⊢ (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = (𝑥 ∩ {𝐵, 𝐶, 𝐴})
7	6	neeq1i 2858	. . . . . . 7 ⊢ ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅ ↔ (𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅)
8	7	bicomi 214	. . . . . 6 ⊢ ((𝑥 ∩ {𝐵, 𝐶, 𝐴}) ≠ ∅ ↔ (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
9	4, 8	syl8ib 246	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐵 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
10		jao 534	. . . . 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 445	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
13		en3lplem1 8511	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → (𝑥 = 𝐶 → (𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅))
14	13	3coml 1272	. . . . . 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 4284	. . . . . . 7 ⊢ {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
17	16	ineq2i 3811	. . . . . 6 ⊢ (𝑥 ∩ {𝐶, 𝐴, 𝐵}) = (𝑥 ∩ {𝐴, 𝐵, 𝐶})
18	17	neeq1i 2858	. . . . 5 ⊢ ((𝑥 ∩ {𝐶, 𝐴, 𝐵}) ≠ ∅ ↔ (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
19	15, 18	syl8ib 246	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐶 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)))
20	19	imp 445	. . 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 4231	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}
23	22	eleq2i 2693	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)})
24	21, 23	syl6ib 241	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}))
25		abid 2610	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶))
26	24, 25	syl6ib 241	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)))
27		df-3or 1038	. . . . 5 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) ↔ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
28	26, 27	syl6ib 241	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∨ 𝑥 = 𝐶)))
29	28	imp 445	. . 3 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) ∨ 𝑥 = 𝐶))
30	12, 20, 29	mpjaod 396	. 2 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ {𝐴, 𝐵, 𝐶}) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)
31	30	ex 450	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794   ∩ cin 3573  ∅c0 3915  {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-nul 3916  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  en3lp  8513