Theorem kmlem12 8866

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 27-Mar-2004.)

Hypothesis

Ref	Expression
kmlem9.1	⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}

Assertion

Ref	Expression
kmlem12	⊢ (∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ → (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑣,𝑢,𝑡   𝑦,𝐴,𝑧,𝑣

Allowed substitution hints:   𝐴(𝑥,𝑢,𝑡)

Proof of Theorem kmlem12

Step	Hyp	Ref	Expression
1		kmlem9.1	. . . . 5 ⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}
2	1	raleqi 3119	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑧 ∈ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
3		df-ral 2901	. . . 4 ⊢ (∀𝑧 ∈ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑧(𝑧 ∈ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))))
4		vex 3176	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5		eqeq1 2614	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
6	5	rexbidv 3034	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → (∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ ∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
7	4, 6	elab 3319	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ↔ ∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
8	7	imbi1i 338	. . . . . . 7 ⊢ ((𝑧 ∈ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ↔ (∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))))
9		r19.23v 3005	. . . . . . 7 ⊢ (∀𝑡 ∈ 𝑥 (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ↔ (∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))))
10	8, 9	bitr4i 266	. . . . . 6 ⊢ ((𝑧 ∈ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ↔ ∀𝑡 ∈ 𝑥 (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))))
11	10	albii 1737	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ↔ ∀𝑧∀𝑡 ∈ 𝑥 (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))))
12		ralcom4 3197	. . . . 5 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑧(𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ↔ ∀𝑧∀𝑡 ∈ 𝑥 (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))))
13		vex 3176	. . . . . . . 8 ⊢ 𝑡 ∈ V
14		difexg 4735	. . . . . . . 8 ⊢ (𝑡 ∈ V → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∈ V)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∈ V
16		neeq1 2844	. . . . . . . 8 ⊢ (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ ↔ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅))
17		ineq1 3769	. . . . . . . . . 10 ⊢ (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ∩ 𝑦) = ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦))
18	17	eleq2d 2673	. . . . . . . . 9 ⊢ (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
19	18	eubidv 2478	. . . . . . . 8 ⊢ (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
20	16, 19	imbi12d 333	. . . . . . 7 ⊢ (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → ((𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦))))
21	15, 20	ceqsalv 3206	. . . . . 6 ⊢ (∀𝑧(𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ↔ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
22	21	ralbii 2963	. . . . 5 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑧(𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ↔ ∀𝑡 ∈ 𝑥 ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
23	11, 12, 22	3bitr2i 287	. . . 4 ⊢ (∀𝑧(𝑧 ∈ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ↔ ∀𝑡 ∈ 𝑥 ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
24	2, 3, 23	3bitri 285	. . 3 ⊢ (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑡 ∈ 𝑥 ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
25		ralim 2932	. . 3 ⊢ (∀𝑡 ∈ 𝑥 ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)) → (∀𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
26	24, 25	sylbi 206	. 2 ⊢ (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → (∀𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
27		difeq1 3683	. . . . . . . 8 ⊢ (𝑡 = 𝑧 → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑡})))
28		sneq 4135	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑧 → {𝑡} = {𝑧})
29	28	difeq2d 3690	. . . . . . . . . 10 ⊢ (𝑡 = 𝑧 → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
30	29	unieqd 4382	. . . . . . . . 9 ⊢ (𝑡 = 𝑧 → ∪ (𝑥 ∖ {𝑡}) = ∪ (𝑥 ∖ {𝑧}))
31	30	difeq2d 3690	. . . . . . . 8 ⊢ (𝑡 = 𝑧 → (𝑧 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
32	27, 31	eqtrd 2644	. . . . . . 7 ⊢ (𝑡 = 𝑧 → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
33	32	neeq1d 2841	. . . . . 6 ⊢ (𝑡 = 𝑧 → ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ ↔ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅))
34	33	cbvralv 3147	. . . . 5 ⊢ (∀𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ ↔ ∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅)
35	32	ineq1d 3775	. . . . . . . 8 ⊢ (𝑡 = 𝑧 → ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦) = ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦))
36	35	eleq2d 2673	. . . . . . 7 ⊢ (𝑡 = 𝑧 → (𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦)))
37	36	eubidv 2478	. . . . . 6 ⊢ (𝑡 = 𝑧 → (∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦)))
38	37	cbvralv 3147	. . . . 5 ⊢ (∀𝑡 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦))
39	34, 38	imbi12i 339	. . . 4 ⊢ ((∀𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)) ↔ (∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦)))
40		in12 3786	. . . . . . . . . . 11 ⊢ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)) = (𝑦 ∩ (𝑧 ∩ ∪ 𝐴))
41		incom 3767	. . . . . . . . . . 11 ⊢ (𝑦 ∩ (𝑧 ∩ ∪ 𝐴)) = ((𝑧 ∩ ∪ 𝐴) ∩ 𝑦)
42	40, 41	eqtri 2632	. . . . . . . . . 10 ⊢ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)) = ((𝑧 ∩ ∪ 𝐴) ∩ 𝑦)
43	1	kmlem11 8865	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ ∪ 𝐴) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
44	43	ineq1d 3775	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ∩ ∪ 𝐴) ∩ 𝑦) = ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦))
45	42, 44	syl5req 2657	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦) = (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)))
46	45	eleq2d 2673	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → (𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴))))
47	46	eubidv 2478	. . . . . . 7 ⊢ (𝑧 ∈ 𝑥 → (∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴))))
48		ax-1 6	. . . . . . 7 ⊢ (∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴))))
49	47, 48	syl6bi 242	. . . . . 6 ⊢ (𝑧 ∈ 𝑥 → (∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)))))
50	49	ralimia 2934	. . . . 5 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴))))
51	50	imim2i 16	. . . 4 ⊢ ((∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦)) → (∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)))))
52	39, 51	sylbi 206	. . 3 ⊢ ((∀𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)) → (∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)))))
53	52	com12 32	. 2 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ → ((∀𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)))))
54	26, 53	syl5 33	1 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ → (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)))))

Syntax hints:   → wi 4  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539  ∅c0 3874  {csn 4125  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-uni 4373  df-iun 4457

This theorem is referenced by:  kmlem13  8867