Theorem kmlem12 10168

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		difeq1 4070	. . . . . . 7 ⊢ (𝑡 = 𝑧 → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑡})))
2		sneq 4597	. . . . . . . . . 10 ⊢ (𝑡 = 𝑧 → {𝑡} = {𝑧})
3	2	difeq2d 4077	. . . . . . . . 9 ⊢ (𝑡 = 𝑧 → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {𝑧}))
4	3	unieqd 4883	. . . . . . . 8 ⊢ (𝑡 = 𝑧 → ∪ (𝑥 ∖ {𝑡}) = ∪ (𝑥 ∖ {𝑧}))
5	4	difeq2d 4077	. . . . . . 7 ⊢ (𝑡 = 𝑧 → (𝑧 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
6	1, 5	eqtrd 2797	. . . . . 6 ⊢ (𝑡 = 𝑧 → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
7	6	neeq1d 3016	. . . . 5 ⊢ (𝑡 = 𝑧 → ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ ↔ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅))
8	7	cbvralvw 3242	. . . 4 ⊢ (∀𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ ↔ ∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅)
9	6	ineq1d 4168	. . . . . . 7 ⊢ (𝑡 = 𝑧 → ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦) = ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦))
10	9	eleq2d 2848	. . . . . 6 ⊢ (𝑡 = 𝑧 → (𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦)))
11	10	eubidv 2613	. . . . 5 ⊢ (𝑡 = 𝑧 → (∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦)))
12	11	cbvralvw 3242	. . . 4 ⊢ (∀𝑡 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦))
13	8, 12	imbi12i 353	. . 3 ⊢ ((∀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)))
14		in12 4177	. . . . . . . . . 10 ⊢ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)) = (𝑦 ∩ (𝑧 ∩ ∪ 𝐴))
15		incom 4158	. . . . . . . . . 10 ⊢ (𝑦 ∩ (𝑧 ∩ ∪ 𝐴)) = ((𝑧 ∩ ∪ 𝐴) ∩ 𝑦)
16	14, 15	eqtri 2785	. . . . . . . . 9 ⊢ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)) = ((𝑧 ∩ ∪ 𝐴) ∩ 𝑦)
17		kmlem9.1	. . . . . . . . . . 11 ⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}
18	17	kmlem11 10167	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ ∪ 𝐴) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})))
19	18	ineq1d 4168	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ∩ ∪ 𝐴) ∩ 𝑦) = ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦))
20	16, 19	eqtr2id 2810	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦) = (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)))
21	20	eleq2d 2848	. . . . . . 7 ⊢ (𝑧 ∈ 𝑥 → (𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴))))
22	21	eubidv 2613	. . . . . 6 ⊢ (𝑧 ∈ 𝑥 → (∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴))))
23		ax-1 6	. . . . . 6 ⊢ (∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴))))
24	22, 23	biimtrdi 256	. . . . 5 ⊢ (𝑧 ∈ 𝑥 → (∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)))))
25	24	ralimia 3098	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑦) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴))))
26	25	imim2i 17	. . 3 ⊢ ((∀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)))))
27	13, 26	sylbi 220	. 2 ⊢ ((∀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)))))
28	17	raleqi 3319	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑧 ∈ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
29		df-ral 3079	. . . 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))))
30		vex 3457	. . . . . . . . 9 ⊢ 𝑧 ∈ V
31		eqeq1 2766	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
32	31	rexbidv 3188	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → (∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ ∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
33	30, 32	elab 3636	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ↔ ∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
34	33	imbi1i 352	. . . . . . 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))))
35		r19.23v 3191	. . . . . . 7 ⊢ (∀𝑡 ∈ 𝑥 (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ↔ (∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))))
36	34, 35	bitr4i 281	. . . . . 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))))
37	36	albii 1852	. . . . 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))))
38		ralcom4 3290	. . . . 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))))
39		vex 3457	. . . . . . . 8 ⊢ 𝑡 ∈ V
40	39	difexi 5299	. . . . . . 7 ⊢ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∈ V
41		neeq1 3019	. . . . . . . 8 ⊢ (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ ↔ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅))
42		ineq1 4162	. . . . . . . . . 10 ⊢ (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ∩ 𝑦) = ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦))
43	42	eleq2d 2848	. . . . . . . . 9 ⊢ (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
44	43	eubidv 2613	. . . . . . . 8 ⊢ (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
45	41, 44	imbi12d 347	. . . . . . 7 ⊢ (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → ((𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦))))
46	40, 45	ceqsalv 3492	. . . . . 6 ⊢ (∀𝑧(𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ↔ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
47	46	ralbii 3110	. . . . 5 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑧(𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) → (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))) ↔ ∀𝑡 ∈ 𝑥 ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
48	37, 38, 47	3bitr2i 302	. . . 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)))
49	28, 29, 48	3bitri 300	. . 3 ⊢ (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑡 ∈ 𝑥 ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
50		ralim 3104	. . 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)))
51	49, 50	sylbi 220	. 2 ⊢ (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → (∀𝑡 ∈ 𝑥 (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ≠ ∅ → ∀𝑡 ∈ 𝑥 ∃!𝑣 𝑣 ∈ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ 𝑦)))
52	27, 51	syl11 34	1 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ → (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴)))))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃!weu 2595  {cab 2740   ≠ wne 2957  ∀wral 3078  ∃wrex 3088   ∖ cdif 3899   ∩ cin 3901  ∅c0 4282  {csn 4587  ∪ cuni 4870

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2734  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-sn 4588  df-uni 4871  df-iun 4956

This theorem is used by:  kmlem13  10169