Theorem hauspwpwf1 21791

Description: Lemma for hauspwpwdom 21792. 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 3834	. . . . . . . . . 10 ⊢ (𝑗 ∩ 𝐴) ⊆ 𝐴
2		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑗 ∈ V
3	2	inex1 4799	. . . . . . . . . . 11 ⊢ (𝑗 ∩ 𝐴) ∈ V
4	3	elpw 4164	. . . . . . . . . 10 ⊢ ((𝑗 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑗 ∩ 𝐴) ⊆ 𝐴)
5	1, 4	mpbir 221	. . . . . . . . 9 ⊢ (𝑗 ∩ 𝐴) ∈ 𝒫 𝐴
6		eleq1 2689	. . . . . . . . 9 ⊢ (𝑎 = (𝑗 ∩ 𝐴) → (𝑎 ∈ 𝒫 𝐴 ↔ (𝑗 ∩ 𝐴) ∈ 𝒫 𝐴))
7	5, 6	mpbiri 248	. . . . . . . 8 ⊢ (𝑎 = (𝑗 ∩ 𝐴) → 𝑎 ∈ 𝒫 𝐴)
8	7	adantl 482	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) → 𝑎 ∈ 𝒫 𝐴)
9	8	rexlimivw 3029	. . . . . 6 ⊢ (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) → 𝑎 ∈ 𝒫 𝐴)
10	9	abssi 3677	. . . . 5 ⊢ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ⊆ 𝒫 𝐴
11		haustop 21135	. . . . . . . . 9 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
12		hauspwpwf1.x	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
13	12	topopn 20711	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
14	11, 13	syl 17	. . . . . . . 8 ⊢ (𝐽 ∈ Haus → 𝑋 ∈ 𝐽)
15		ssexg 4804	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
16	14, 15	sylan2 491	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐽 ∈ Haus) → 𝐴 ∈ V)
17	16	ancoms 469	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
18		pwexg 4850	. . . . . 6 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
19		elpw2g 4827	. . . . . 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 248	. . . 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 798	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝐽 ∈ Haus)
24	12	clsss3 20863	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
25	11, 24	sylan 488	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
26	25	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
27		simplrl 800	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
28	26, 27	sseldd 3604	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ 𝑋)
29		simplrr 801	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ ((cls‘𝐽)‘𝐴))
30	26, 29	sseldd 3604	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ 𝑋)
31		simpr 477	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) ∧ 𝑥 ≠ 𝑦) → 𝑥 ≠ 𝑦)
32	12	hausnei 21132	. . . . . . . . 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 1328	. . . . . . . 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 802	. . . . . . . . . . . . 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 1109	. . . . . . . . . . . . 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 2623	. . . . . . . . . . . . 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 2004	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝑥 ∈ 𝑗 ↔ 𝑥 ∈ 𝑘))
38		ineq1 3807	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (𝑗 ∩ 𝐴) = (𝑘 ∩ 𝐴))
39	38	eqeq2d 2632	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → ((𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴) ↔ (𝑘 ∩ 𝐴) = (𝑘 ∩ 𝐴)))
40	37, 39	anbi12d 747	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑘 ∧ (𝑘 ∩ 𝐴) = (𝑘 ∩ 𝐴))))
41	40	rspcev 3309	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ 𝐽 ∧ (𝑥 ∈ 𝑘 ∧ (𝑘 ∩ 𝐴) = (𝑘 ∩ 𝐴))) → ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
42	34, 35, 36, 41	syl12anc 1324	. . . . . . . . . . . 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 3203	. . . . . . . . . . . . . 14 ⊢ 𝑘 ∈ V
44	43	inex1 4799	. . . . . . . . . . . . 13 ⊢ (𝑘 ∩ 𝐴) ∈ V
45		eqeq1 2626	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝑘 ∩ 𝐴) → (𝑎 = (𝑗 ∩ 𝐴) ↔ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
46	45	anbi2d 740	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑘 ∩ 𝐴) → ((𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))))
47	46	rexbidv 3052	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑘 ∩ 𝐴) → (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))))
48	44, 47	elab 3350	. . . . . . . . . . . 12 ⊢ ((𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ↔ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
49	42, 48	sylibr 224	. . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ Top)
51	50	ad3antrrr 766	. . . . . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝐴 ⊆ 𝑋)
53	52	ad3antrrr 766	. . . . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((cls‘𝐽)‘𝐴))) → 𝑦 ∈ ((cls‘𝐽)‘𝐴))
55	54	ad3antrrr 766	. . . . . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅)) → 𝑙 ∈ 𝐽)
57	56	ad2antlr 763	. . . . . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . . . 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 20704	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ Top ∧ 𝑙 ∈ 𝐽 ∧ 𝑗 ∈ 𝐽) → (𝑙 ∩ 𝑗) ∈ 𝐽)
60	51, 57, 58, 59	syl3anc 1326	. . . . . . . . . . . . . . . . . 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 1068	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅)) → 𝑦 ∈ 𝑙)
62	61	ad2antlr 763	. . . . . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . . . 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 3798	. . . . . . . . . . . . . . . . . 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 20879	. . . . . . . . . . . . . . . . . 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 1334	. . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑙 ∩ 𝑗) ∩ 𝐴) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴))
68	66, 67	sylib 208	. . . . . . . . . . . . . . . 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 3796	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴) ↔ (𝑧 ∈ (𝑙 ∩ 𝑗) ∧ 𝑧 ∈ 𝐴))
70		elin 3796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝑙 ∩ 𝑗) ↔ (𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗))
71	70	anbi1i 731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ (𝑙 ∩ 𝑗) ∧ 𝑧 ∈ 𝐴) ↔ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))
72	69, 71	bitri 264	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝑙 ∩ 𝑗) ∩ 𝐴) ↔ ((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴))
73		elin 3796	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ (𝑗 ∩ 𝐴) ↔ (𝑧 ∈ 𝑗 ∧ 𝑧 ∈ 𝐴))
74	73	biimpri 218	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ 𝑗 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝑗 ∩ 𝐴))
75	74	adantll 750	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝑗 ∩ 𝐴))
76	75	ad2antll 765	. . . . . . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ 𝑙 ∧ 𝑧 ∈ 𝑗) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝑙)
78	77	ad2antll 765	. . . . . . . . . . . . . . . . . . . . . 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 1069	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ 𝐽 ∧ 𝑙 ∈ 𝐽) ∧ (𝑥 ∈ 𝑘 ∧ 𝑦 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅)) → (𝑘 ∩ 𝑙) = ∅)
80	79	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 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 4033	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅) → ¬ 𝑧 ∈ 𝑘)
82		inss1 3833	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∩ 𝐴) ⊆ 𝑘
83	82	sseli 3599	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (𝑘 ∩ 𝐴) → 𝑧 ∈ 𝑘)
84	81, 83	nsyl 135	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ 𝑙 ∧ (𝑘 ∩ 𝑙) = ∅) → ¬ 𝑧 ∈ (𝑘 ∩ 𝐴))
85	78, 80, 84	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 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))
86		nelneq2 2726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ (𝑗 ∩ 𝐴) ∧ ¬ 𝑧 ∈ (𝑘 ∩ 𝐴)) → ¬ (𝑗 ∩ 𝐴) = (𝑘 ∩ 𝐴))
87	76, 85, 86	syl2anc 693	. . . . . . . . . . . . . . . . . . . 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))
88		eqcom 2629	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∩ 𝐴) = (𝑘 ∩ 𝐴) ↔ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))
89	87, 88	sylnib 318	. . . . . . . . . . . . . . . . . . 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))
90	89	expr 643	. . . . . . . . . . . . . . . . . 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)))
91	72, 90	syl5bi 232	. . . . . . . . . . . . . . . . 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)))
92	91	exlimdv 1861	. . . . . . . . . . . . . . . 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)))
93	68, 92	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))
94	93	anassrs 680	. . . . . . . . . . . . . 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))
95		nan 604	. . . . . . . . . . . . . 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)))
96	94, 95	mpbir 221	. . . . . . . . . . . . 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)))
97	96	nrexdv 3001	. . . . . . . . . . . 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)))
98	45	anbi2d 740	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑘 ∩ 𝐴) → ((𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))))
99	98	rexbidv 3052	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑘 ∩ 𝐴) → (∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴))))
100	44, 99	elab 3350	. . . . . . . . . . . 12 ⊢ ((𝑘 ∩ 𝐴) ∈ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} ↔ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ (𝑘 ∩ 𝐴) = (𝑗 ∩ 𝐴)))
101	97, 100	sylnibr 319	. . . . . . . . . . 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))})
102		nelne1 2890	. . . . . . . . . . 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))})
103	49, 101, 102	syl2anc 693	. . . . . . . . . 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))})
104	103	expr 643	. . . . . . . . 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))}))
105	104	rexlimdvva 3038	. . . . . . . 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))}))
106	33, 105	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))})
107	106	ex 450	. . . . . 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))}))
108	107	necon4d 2818	. . . . 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))
109		eleq1 2689	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑗 ↔ 𝑦 ∈ 𝑗))
110	109	anbi1d 741	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))))
111	110	rexbidv 3052	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴)) ↔ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))))
112	111	abbidv 2741	. . . . 5 ⊢ (𝑥 = 𝑦 → {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))} = {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})
113	108, 112	impbid1 215	. . . 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))
114	113	ex 450	. . 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)))
115	22, 114	dom2lem 7995	. 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)
116		hauspwpwf1.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})
117		f1eq1 6096	. . 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))
118	116, 117	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)
119	115, 118	sylibr 224	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∃wrex 2913  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436   ↦ cmpt 4729  –1-1→wf1 5885  ‘cfv 5888  Topctop 20698  clsccl 20822  Hauscha 21112

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-cld 20823  df-ntr 20824  df-cls 20825  df-haus 21119

This theorem is referenced by:  hauspwpwdom  21792