Theorem iscnrm3rlem8 50055

Description: Lemma for iscnrm3r 50056. Disjoint open neighborhoods in the subspace topology are disjoint open neighborhoods in the original topology given that the subspace is an open set in the original topology. Therefore, given any two sets separated in the original topology and separated by open neighborhoods in the subspace topology, they must be separated by open neighborhoods in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)

Assertion

Ref	Expression
iscnrm3rlem8	⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → (∃𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))

Distinct variable groups:   𝑘,𝐽,𝑙,𝑚,𝑛   𝑆,𝑘,𝑙,𝑚,𝑛   𝑇,𝑘,𝑙,𝑚,𝑛

Proof of Theorem iscnrm3rlem8

Step	Hyp	Ref	Expression
1		sseq2 3934	. . 3 ⊢ (𝑛 = 𝑙 → (𝑆 ⊆ 𝑛 ↔ 𝑆 ⊆ 𝑙))
2		ineq1 4110	. . . 4 ⊢ (𝑛 = 𝑙 → (𝑛 ∩ 𝑚) = (𝑙 ∩ 𝑚))
3	2	eqeq1d 2762	. . 3 ⊢ (𝑛 = 𝑙 → ((𝑛 ∩ 𝑚) = ∅ ↔ (𝑙 ∩ 𝑚) = ∅))
4	1, 3	3anbi13d 1466	. 2 ⊢ (𝑛 = 𝑙 → ((𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ (𝑆 ⊆ 𝑙 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑙 ∩ 𝑚) = ∅)))
5		sseq2 3934	. . 3 ⊢ (𝑚 = 𝑘 → (𝑇 ⊆ 𝑚 ↔ 𝑇 ⊆ 𝑘))
6		ineq2 4111	. . . 4 ⊢ (𝑚 = 𝑘 → (𝑙 ∩ 𝑚) = (𝑙 ∩ 𝑘))
7	6	eqeq1d 2762	. . 3 ⊢ (𝑚 = 𝑘 → ((𝑙 ∩ 𝑚) = ∅ ↔ (𝑙 ∩ 𝑘) = ∅))
8	5, 7	3anbi23d 1467	. 2 ⊢ (𝑚 = 𝑘 → ((𝑆 ⊆ 𝑙 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑙 ∩ 𝑚) = ∅) ↔ (𝑆 ⊆ 𝑙 ∧ 𝑇 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))
9		simp11 1222	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → 𝐽 ∈ Top)
10		simp12l 1305	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → 𝑆 ∈ 𝒫 ∪ 𝐽)
11	10	elpwid 4565	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → 𝑆 ⊆ ∪ 𝐽)
12		simp12r 1306	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → 𝑇 ∈ 𝒫 ∪ 𝐽)
13	12	elpwid 4565	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → 𝑇 ⊆ ∪ 𝐽)
14		simp2l 1218	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → 𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
15	9, 11, 13, 14	iscnrm3rlem7 50054	. . 3 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → 𝑙 ∈ 𝐽)
16		simp2r 1219	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
17	9, 11, 13, 16	iscnrm3rlem7 50054	. . 3 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → 𝑘 ∈ 𝐽)
18		simp13l 1307	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
19		simp31 1228	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙)
20	9, 11, 18, 19	iscnrm3rlem4 50051	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → 𝑆 ⊆ 𝑙)
21		incom 4106	. . . . . 6 ⊢ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = (𝑇 ∩ ((cls‘𝐽)‘𝑆))
22		simp13r 1308	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
23	21, 22	eqtr3id 2809	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → (𝑇 ∩ ((cls‘𝐽)‘𝑆)) = ∅)
24		simp32 1229	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘)
25	9, 13, 23, 24	iscnrm3rlem4 50051	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → 𝑇 ⊆ 𝑘)
26		simp33 1230	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → (𝑙 ∩ 𝑘) = ∅)
27	20, 25, 26	3jca 1146	. . 3 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → (𝑆 ⊆ 𝑙 ∧ 𝑇 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅))
28	15, 17, 27	3jca 1146	. 2 ⊢ (((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) ∧ (𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ∧ 𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → (𝑙 ∈ 𝐽 ∧ 𝑘 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑙 ∧ 𝑇 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))
29	4, 8, 28	iscnrm3lem7 50047	1 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → (∃𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278  𝒫 cpw 4556  ∪ cuni 4866  ‘cfv 6538  (class class class)co 7420   ↾_t crest 17591  Topctop 23211  clsccl 23336

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-en 8974  df-fin 8977  df-fi 9403  df-rest 17593  df-topgen 17614  df-top 23212  df-topon 23229  df-bases 23264  df-cld 23337  df-cls 23339

This theorem is used by:  iscnrm3r  50056