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Theorem iscnrm3r 47793
Description: Lemma for iscnrm3 47797. If all subspaces of a topology are normal, i.e., two disjoint closed sets can be separated by open neighborhoods, then in the original topology two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
Assertion
Ref Expression
iscnrm3r (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 𝐽𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) → ((𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽) → (((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)))))
Distinct variable groups:   𝑘,𝐽,𝑙,𝑚,𝑛   𝑆,𝑘,𝑙,𝑚,𝑛   𝑇,𝑘,𝑙,𝑚,𝑛   𝐽,𝑐,𝑑,𝑧,𝑘,𝑙   𝑆,𝑐,𝑑,𝑧   𝑇,𝑐,𝑑,𝑧

Proof of Theorem iscnrm3r
StepHypRef Expression
1 oveq2 7410 . . . . . . 7 (𝑧 = ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) → (𝐽t 𝑧) = (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
21fveq2d 6886 . . . . . 6 (𝑧 = ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) → (Clsd‘(𝐽t 𝑧)) = (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))))
31rexeqdv 3318 . . . . . . . . 9 (𝑧 = ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) → (∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅) ↔ ∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)))
41, 3rexeqbidv 3335 . . . . . . . 8 (𝑧 = ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) → (∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅) ↔ ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)))
54imbi2d 340 . . . . . . 7 (𝑧 = ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) → (((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) ↔ ((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅))))
62, 5raleqbidv 3334 . . . . . 6 (𝑧 = ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) → (∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) ↔ ∀𝑑 ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅))))
72, 6raleqbidv 3334 . . . . 5 (𝑧 = ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) → (∀𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) ↔ ∀𝑐 ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))∀𝑑 ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅))))
87rspcv 3600 . . . 4 (( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 𝐽 → (∀𝑧 ∈ 𝒫 𝐽𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) → ∀𝑐 ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))∀𝑑 ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅))))
983ad2ant1 1130 . . 3 ((( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))) → (∀𝑧 ∈ 𝒫 𝐽𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) → ∀𝑐 ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))∀𝑑 ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅))))
10 ineq12 4200 . . . . . . 7 ((𝑐 = (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∧ 𝑑 = (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) → (𝑐𝑑) = ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∩ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))))
1110eqeq1d 2726 . . . . . 6 ((𝑐 = (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∧ 𝑑 = (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) → ((𝑐𝑑) = ∅ ↔ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∩ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) = ∅))
12 simpl 482 . . . . . . . . 9 ((𝑐 = (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∧ 𝑑 = (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) → 𝑐 = (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)))
1312sseq1d 4006 . . . . . . . 8 ((𝑐 = (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∧ 𝑑 = (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) → (𝑐𝑙 ↔ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙))
14 simpr 484 . . . . . . . . 9 ((𝑐 = (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∧ 𝑑 = (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) → 𝑑 = (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)))
1514sseq1d 4006 . . . . . . . 8 ((𝑐 = (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∧ 𝑑 = (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) → (𝑑𝑘 ↔ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘))
1613, 153anbi12d 1433 . . . . . . 7 ((𝑐 = (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∧ 𝑑 = (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) → ((𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅) ↔ ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙𝑘) = ∅)))
17162rexbidv 3211 . . . . . 6 ((𝑐 = (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∧ 𝑑 = (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) → (∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅) ↔ ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙𝑘) = ∅)))
1811, 17imbi12d 344 . . . . 5 ((𝑐 = (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∧ 𝑑 = (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) → (((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) ↔ (((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∩ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙𝑘) = ∅))))
1918rspc2gv 3614 . . . 4 (((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))) → (∀𝑐 ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))∀𝑑 ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) → (((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∩ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙𝑘) = ∅))))
20193adant1 1127 . . 3 ((( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))) → (∀𝑐 ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))∀𝑑 ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) → (((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∩ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙𝑘) = ∅))))
219, 20syld 47 . 2 ((( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))) → (∀𝑧 ∈ 𝒫 𝐽𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) → (((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∩ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙𝑘) = ∅))))
22 iscnrm3rlem3 47787 . . 3 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽)) → (( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))))
23223adant3 1129 . 2 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → (( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))))
24 disjdifb 47706 . . . 4 ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∩ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) = ∅
2524a1i 11 . . 3 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → ((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∩ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) = ∅)
26 iscnrm3rlem8 47792 . . 3 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → (∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙𝑘) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)))
2725, 26embantd 59 . 2 ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → ((((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∩ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆))) = ∅ → ∃𝑙 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙𝑘) = ∅)) → ∃𝑛𝐽𝑚𝐽 (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)))
2821, 23, 27iscnrm3lem4 47781 1 (𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 𝐽𝑐 ∈ (Clsd‘(𝐽t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽t 𝑧))((𝑐𝑑) = ∅ → ∃𝑙 ∈ (𝐽t 𝑧)∃𝑘 ∈ (𝐽t 𝑧)(𝑐𝑙𝑑𝑘 ∧ (𝑙𝑘) = ∅)) → ((𝑆 ∈ 𝒫 𝐽𝑇 ∈ 𝒫 𝐽) → (((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑆𝑛𝑇𝑚 ∧ (𝑛𝑚) = ∅)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  wrex 3062  cdif 3938  cin 3940  wss 3941  c0 4315  𝒫 cpw 4595   cuni 4900  cfv 6534  (class class class)co 7402  t crest 17367  Topctop 22719  Clsdccld 22844  clsccl 22846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-en 8937  df-fin 8940  df-fi 9403  df-rest 17369  df-topgen 17390  df-top 22720  df-topon 22737  df-bases 22773  df-cld 22847  df-cls 22849
This theorem is referenced by:  iscnrm3  47797
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