Theorem iscnrm3rlem8 47879

Description: Lemma for iscnrm3r 47880. 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 4004	. . 3 ⊢ (𝑛 = 𝑙 → (𝑆 ⊆ 𝑛 ↔ 𝑆 ⊆ 𝑙))
2		ineq1 4201	. . . 4 ⊢ (𝑛 = 𝑙 → (𝑛 ∩ 𝑚) = (𝑙 ∩ 𝑚))
3	2	eqeq1d 2729	. . 3 ⊢ (𝑛 = 𝑙 → ((𝑛 ∩ 𝑚) = ∅ ↔ (𝑙 ∩ 𝑚) = ∅))
4	1, 3	3anbi13d 1435	. 2 ⊢ (𝑛 = 𝑙 → ((𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ (𝑆 ⊆ 𝑙 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑙 ∩ 𝑚) = ∅)))
5		sseq2 4004	. . 3 ⊢ (𝑚 = 𝑘 → (𝑇 ⊆ 𝑚 ↔ 𝑇 ⊆ 𝑘))
6		ineq2 4202	. . . 4 ⊢ (𝑚 = 𝑘 → (𝑙 ∩ 𝑚) = (𝑙 ∩ 𝑘))
7	6	eqeq1d 2729	. . 3 ⊢ (𝑚 = 𝑘 → ((𝑙 ∩ 𝑚) = ∅ ↔ (𝑙 ∩ 𝑘) = ∅))
8	5, 7	3anbi23d 1436	. 2 ⊢ (𝑚 = 𝑘 → ((𝑆 ⊆ 𝑙 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑙 ∩ 𝑚) = ∅) ↔ (𝑆 ⊆ 𝑙 ∧ 𝑇 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))
9		simp11 1201	. . . 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 1284	. . . . 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 4607	. . . 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 1285	. . . . 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 4607	. . . 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 1197	. . . 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 47878	. . 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 1198	. . . 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 47878	. . 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 1286	. . . . 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 1207	. . . . 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 47875	. . . 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 4197	. . . . . 6 ⊢ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = (𝑇 ∩ ((cls‘𝐽)‘𝑆))
22		simp13r 1287	. . . . . 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 2781	. . . . 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 1208	. . . . 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 47875	. . . 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 1209	. . . 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 1126	. . 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 1126	. 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 47871	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) = ∅)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∃wrex 3065   ∖ cdif 3941   ∩ cin 3943   ⊆ wss 3944  ∅c0 4318  𝒫 cpw 4598  ∪ cuni 4903  ‘cfv 6542  (class class class)co 7414   ↾_t crest 17387  Topctop 22769  clsccl 22896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-en 8954  df-fin 8957  df-fi 9420  df-rest 17389  df-topgen 17410  df-top 22770  df-topon 22787  df-bases 22823  df-cld 22897  df-cls 22899

This theorem is referenced by:  iscnrm3r  47880