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Theorem iscnrm3r 47852
Description: Lemma for iscnrm3 47856. 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 7413 . . . . . . 7 (𝑧 = ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) → (𝐽t 𝑧) = (𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))
21fveq2d 6889 . . . . . 6 (𝑧 = ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) → (Clsd‘(𝐽t 𝑧)) = (Clsd‘(𝐽t ( 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))))
31rexeqdv 3320 . . . . . . . . 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 3337 . . . . . . . 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 3336 . . . . . 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 3336 . . . . 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 3602 . . . 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 4202 . . . . . . 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 2728 . . . . . 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 4008 . . . . . . . 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 4008 . . . . . . . 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 3213 . . . . . 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 3616 . . . 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 47846 . . 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 47765 . . . 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 47851 . . 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 47840 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 3055  wrex 3064  cdif 3940  cin 3942  wss 3943  c0 4317  𝒫 cpw 4597   cuni 4902  cfv 6537  (class class class)co 7405  t crest 17375  Topctop 22750  Clsdccld 22875  clsccl 22877
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17377  df-topgen 17398  df-top 22751  df-topon 22768  df-bases 22804  df-cld 22878  df-cls 22880
This theorem is referenced by:  iscnrm3  47856
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