Step |
Hyp |
Ref |
Expression |
1 |
|
iscnrm3 |
⊢ ( 𝐽 ∈ CNrm ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 ∪ 𝐽 ∀ 𝑡 ∈ 𝒫 ∪ 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
2 |
|
id |
⊢ ( 𝐽 ∈ Top → 𝐽 ∈ Top ) |
3 |
2
|
sepnsepo |
⊢ ( 𝐽 ∈ Top → ( ∃ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑠 ) ∃ 𝑚 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑡 ) ( 𝑛 ∩ 𝑚 ) = ∅ ↔ ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) |
4 |
3
|
imbi2d |
⊢ ( 𝐽 ∈ Top → ( ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑠 ) ∃ 𝑚 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑡 ) ( 𝑛 ∩ 𝑚 ) = ∅ ) ↔ ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
5 |
4
|
2ralbidv |
⊢ ( 𝐽 ∈ Top → ( ∀ 𝑠 ∈ 𝒫 ∪ 𝐽 ∀ 𝑡 ∈ 𝒫 ∪ 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑠 ) ∃ 𝑚 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑡 ) ( 𝑛 ∩ 𝑚 ) = ∅ ) ↔ ∀ 𝑠 ∈ 𝒫 ∪ 𝐽 ∀ 𝑡 ∈ 𝒫 ∪ 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
6 |
5
|
pm5.32i |
⊢ ( ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 ∪ 𝐽 ∀ 𝑡 ∈ 𝒫 ∪ 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑠 ) ∃ 𝑚 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑡 ) ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 ∪ 𝐽 ∀ 𝑡 ∈ 𝒫 ∪ 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
7 |
1 6
|
bitr4i |
⊢ ( 𝐽 ∈ CNrm ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 ∪ 𝐽 ∀ 𝑡 ∈ 𝒫 ∪ 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑠 ) ∃ 𝑚 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑡 ) ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) |