Theorem hauspwpwf1 23712

Description: Lemma for hauspwpwdom 23713. Points in the closure of a set in a Hausdorff space are characterized by the open neighborhoods they extend into the generating set. (Contributed by Mario Carneiro, 28-Jul-2015.)

Hypotheses

Ref	Expression
hauspwpwf1.x	⊢ 𝑋 = ∪ 𝐽
hauspwpwf1.f	⊢ 𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})

Assertion

Ref	Expression
hauspwpwf1	⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)

Distinct variable groups:   𝑗,𝑎,𝑥,𝐴   𝐽,𝑎,𝑗,𝑥   𝑗,𝑋,𝑥

Allowed substitution hints:   𝐹(𝑥,𝑗,𝑎)   𝑋(𝑎)

Proof of Theorem hauspwpwf1

Dummy variables 𝑘 𝑙 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 4229	. . . . . . . . . 10 ⊢ (𝑗 ∩ 𝐴) ⊆ 𝐴
2		vex 3477	. . . . . . . . . . . 12 ⊢ 𝑗 ∈ V
3	2	inex1 5317	. . . . . . . . . . 11 ⊢ (𝑗 ∩ 𝐴) ∈ V
4	3	elpw 4606	. . . . . . . . . 10 ⊢ ((𝑗 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑗 ∩ 𝐴) ⊆ 𝐴)
5	1, 4	mpbir 230	. . . . . . . . 9 ⊢ (𝑗 ∩ 𝐴) ∈ 𝒫 𝐴
6		eleq1 2820	. . . . . . . . 9 ⊢ (𝑎 = (𝑗 ∩ 𝐴) → (𝑎 ∈ 𝒫 𝐴 ↔ (𝑗 ∩ 𝐴) ∈ 𝒫 𝐴))
7	5, 6	mpbiri 258	. . . . . . . 8 ⊢ (𝑎 = (𝑗 ∩ 𝐴) → 𝑎 ∈ 𝒫 𝐴)
8	7	adantl 481	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) → 𝑎 ∈ 𝒫 𝐴)
9	8	rexlimivw 3150	. . . . . 6 ⊢ (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) → 𝑎 ∈ 𝒫 𝐴)
10	9	abssi 4067	. . . . 5 ⊢ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ⊆ 𝒫 𝐴
11		haustop 23056	. . . . . . . . 9 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
12		hauspwpwf1.x	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
13	12	topopn 22629	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
14	11, 13	syl 17	. . . . . . . 8 ⊢ (𝐽 ∈ Haus → 𝑋 ∈ 𝐽)
15		ssexg 5323	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
16	14, 15	sylan2 592	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐽 ∈ Haus) → 𝐴 ∈ V)
17	16	ancoms 458	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
18		pwexg 5376	. . . . . 6 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
19		elpw2g 5344	. . . . . 6 ⊢ (𝒫 𝐴 ∈ V → ({𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ∈ 𝒫 𝒫 𝐴 ↔ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ⊆ 𝒫 𝐴))
20	17, 18, 19	3syl 18	. . . . 5 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ({𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ∈ 𝒫 𝒫 𝐴 ↔ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ⊆ 𝒫 𝐴))
21	10, 20	mpbiri 258	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ∈ 𝒫 𝒫 𝐴)
22	21	a1d 25	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ∈ 𝒫 𝒫 𝐴))
23		simplll 772	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝐽 ∈ Haus)
24	12	clsss3 22784	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
25	11, 24	sylan 579	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
26	25	ad2antrr 723	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
27		simplrl 774	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
28	26, 27	sseldd 3983	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ 𝑋)
29		simplrr 775	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ ((cls‘𝐽)‘𝐴))
30	26, 29	sseldd 3983	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ 𝑋)
31		simpr 484	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑥 ≠ 𝑦)
32	12	hausnei 23053	. . . . . . . . 9 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑥 ≠ 𝑦)) → ∃𝑘 ∈ 𝐽 ∃𝑙 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))
33	23, 28, 30, 31, 32	syl13anc 1371	. . . . . . . 8 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → ∃𝑘 ∈ 𝐽 ∃𝑙 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))
34		simprll 776	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → 𝑘 ∈ 𝐽)
35		simprr1 1220	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → 𝑥 ∈ 𝑘)
36		eqidd 2732	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → (𝑘 ∩ 𝐴) = (𝑘 ∩ 𝐴))
37		elequ2 2120	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝑥 ∈ 𝑗 ↔ 𝑥 ∈ 𝑘))
38		ineq1 4205	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (𝑗 ∩ 𝐴) = (𝑘 ∩ 𝐴))
39	38	eqeq2d 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → ((𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴) ↔ (𝑘 ∩ 𝐴) = (𝑘 ∩ 𝐴)))
40	37, 39	anbi12d 630	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑘 ∧ (𝑘 ∩ 𝐴) = (𝑘 ∩ 𝐴))))
41	40	rspcev 3612	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ 𝐽 ∧ (𝑥 ∈ 𝑘 ∧ (𝑘 ∩ 𝐴) = (𝑘 ∩ 𝐴))) → ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
42	34, 35, 36, 41	syl12anc 834	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
43		vex 3477	. . . . . . . . . . . . . 14 ⊢ 𝑘 ∈ V
44	43	inex1 5317	. . . . . . . . . . . . 13 ⊢ (𝑘 ∩ 𝐴) ∈ V
45		eqeq1 2735	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝑘 ∩ 𝐴) → (𝑎 = (𝑗 ∩ 𝐴) ↔ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
46	45	anbi2d 628	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑘 ∩ 𝐴) → ((𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))))
47	46	rexbidv 3177	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑘 ∩ 𝐴) → (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))))
48	44, 47	elab 3668	. . . . . . . . . . . 12 ⊢ ((𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ↔ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
49	42, 48	sylibr 233	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → (𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})
50	11	ad2antrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ Top)
51	50	ad3antrrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝐽 ∈ Top)
52		simplr 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝐴 ⊆ 𝑋)
53	52	ad3antrrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝐴 ⊆ 𝑋)
54		simprr 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝑦 ∈ ((cls‘𝐽)‘𝐴))
55	54	ad3antrrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑦 ∈ ((cls‘𝐽)‘𝐴))
56		simplr 766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅)) → 𝑙 ∈ 𝐽)
57	56	ad2antlr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑙 ∈ 𝐽)
58		simprl 768	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑗 ∈ 𝐽)
59		inopn 22622	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ Top ∧ 𝑙 ∈ 𝐽 ∧ 𝑗 ∈ 𝐽) → (𝑙 ∩ 𝑗) ∈ 𝐽)
60	51, 57, 58, 59	syl3anc 1370	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → (𝑙 ∩ 𝑗) ∈ 𝐽)
61		simpr2 1194	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅)) → 𝑦 ∈ 𝑙)
62	61	ad2antlr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑦 ∈ 𝑙)
63		simprr 770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑦 ∈ 𝑗)
64	62, 63	elind 4194	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → 𝑦 ∈ (𝑙 ∩ 𝑗))
65	12	clsndisj 22800	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴)) ∧ ((𝑙 ∩ 𝑗) ∈ 𝐽 ∧ 𝑦 ∈ (𝑙 ∩ 𝑗))) → ((𝑙 ∩ 𝑗) ∩ 𝐴) ≠ ∅)
66	51, 53, 55, 60, 64, 65	syl32anc 1377	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → ((𝑙 ∩ 𝑗) ∩ 𝐴) ≠ ∅)
67		n0 4346	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑙 ∩ 𝑗) ∩ 𝐴) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴))
68	66, 67	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → ∃𝑧 𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴))
69		elin 3964	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴) ↔ (𝑧 ∈ (𝑙 ∩ 𝑗) ∧ 𝑧 ∈ 𝐴))
70		elin 3964	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝑙 ∩ 𝑗) ↔ (𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗))
71	70	anbi1i 623	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ (𝑙 ∩ 𝑗) ∧ 𝑧 ∈ 𝐴) ↔ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))
72	69, 71	bitri 275	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴) ↔ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))
73		elin 3964	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ (𝑗 ∩ 𝐴) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 ∈ 𝐴))
74	73	biimpri 227	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ 𝑗 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝑗 ∩ 𝐴))
75	74	adantll 711	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝑗 ∩ 𝐴))
76	75	ad2antll 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → 𝑧 ∈ (𝑗 ∩ 𝐴))
77		simpll 764	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝑙)
78	77	ad2antll 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → 𝑧 ∈ 𝑙)
79		simpr3 1195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅)) → (𝑘 ∩ 𝑙) = ∅)
80	79	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → (𝑘 ∩ 𝑙) = ∅)
81		minel 4465	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅) → ¬ 𝑧 ∈ 𝑘)
82		elinel1 4195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (𝑘 ∩ 𝐴) → 𝑧 ∈ 𝑘)
83	81, 82	nsyl 140	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅) → ¬ 𝑧 ∈ (𝑘 ∩ 𝐴))
84	78, 80, 83	syl2anc 583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → ¬ 𝑧 ∈ (𝑘 ∩ 𝐴))
85		nelneq2 2857	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ (𝑗 ∩ 𝐴) ∧ ¬ 𝑧 ∈ (𝑘 ∩ 𝐴)) → ¬ (𝑗 ∩ 𝐴) = (𝑘 ∩ 𝐴))
86	76, 84, 85	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → ¬ (𝑗 ∩ 𝐴) = (𝑘 ∩ 𝐴))
87		eqcom 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∩ 𝐴) = (𝑘 ∩ 𝐴) ↔ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))
88	86, 87	sylnib 328	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ ((𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗) ∧ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))
89	88	expr 456	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → (((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
90	72, 89	biimtrid 241	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → (𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
91	90	exlimdv 1935	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → (∃𝑧 𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
92	68, 91	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ (𝑗 ∈ 𝐽 ∧ 𝑦 ∈ 𝑗)) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))
93	92	anassrs 467	. . . . . . . . . . . . . 14 ⊢ (((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ 𝑗 ∈ 𝐽) ∧ 𝑦 ∈ 𝑗) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))
94		nan 827	. . . . . . . . . . . . . 14 ⊢ (((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ 𝑗 ∈ 𝐽) → ¬ (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))) ↔ (((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ 𝑗 ∈ 𝐽) ∧ 𝑦 ∈ 𝑗) → ¬ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
95	93, 94	mpbir 230	. . . . . . . . . . . . 13 ⊢ ((((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) ∧ 𝑗 ∈ 𝐽) → ¬ (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
96	95	nrexdv 3148	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → ¬ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
97	45	anbi2d 628	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑘 ∩ 𝐴) → ((𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))))
98	97	rexbidv 3177	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑘 ∩ 𝐴) → (∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))))
99	44, 98	elab 3668	. . . . . . . . . . . 12 ⊢ ((𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ↔ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
100	96, 99	sylnibr 329	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → ¬ (𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})
101		nelne1 3038	. . . . . . . . . . 11 ⊢ (((𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ∧ ¬ (𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})
102	49, 100, 101	syl2anc 583	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ ((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅))) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})
103	102	expr 456	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) ∧ (𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽)) → ((𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}))
104	103	rexlimdvva 3210	. . . . . . . 8 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → (∃𝑘 ∈ 𝐽 ∃𝑙 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}))
105	33, 104	mpd 15	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})
106	105	ex 412	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → (𝑥 ≠ 𝑦 → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ≠ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}))
107	106	necon4d 2963	. . . . 5 ⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → ({𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} = {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} → 𝑥 = 𝑦))
108		eleq1 2820	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑗 ↔ 𝑦 ∈ 𝑗))
109	108	anbi1d 629	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))))
110	109	rexbidv 3177	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))))
111	110	abbidv 2800	. . . . 5 ⊢ (𝑥 = 𝑦 → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} = {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})
112	107, 111	impbid1 224	. . . 4 ⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → ({𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} = {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ↔ 𝑥 = 𝑦))
113	112	ex 412	. . 3 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴)) → ({𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} = {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ↔ 𝑥 = 𝑦)))
114	22, 113	dom2lem 8992	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)
115		hauspwpwf1.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})
116		f1eq1 6782	. . 3 ⊢ (𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) → (𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴 ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴))
117	115, 116	ax-mp 5	. 2 ⊢ (𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴 ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}):((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)
118	114, 117	sylibr 233	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1780   ∈ wcel 2105  {cab 2708   ≠ wne 2939  ∃wrex 3069  Vcvv 3473   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  ∪ cuni 4908   ↦ cmpt 5231  –1-1→wf1 6540  ‘cfv 6543  Topctop 22616  clsccl 22743  Hauscha 23033

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22617  df-cld 22744  df-ntr 22745  df-cls 22746  df-haus 23040

This theorem is referenced by:  hauspwpwdom  23713