Step |
Hyp |
Ref |
Expression |
1 |
|
restcls2.1 |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
2 |
|
restcls2.2 |
⊢ ( 𝜑 → 𝑋 = ∪ 𝐽 ) |
3 |
|
restcls2.3 |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑋 ) |
4 |
|
restcls2.4 |
⊢ ( 𝜑 → 𝐾 = ( 𝐽 ↾t 𝑌 ) ) |
5 |
|
restcls2.5 |
⊢ ( 𝜑 → 𝑆 ∈ ( Clsd ‘ 𝐾 ) ) |
6 |
|
restclsseplem.6 |
⊢ ( 𝜑 → ( 𝑆 ∩ 𝑇 ) = ∅ ) |
7 |
|
restclssep.7 |
⊢ ( 𝜑 → 𝑇 ∈ ( Clsd ‘ 𝐾 ) ) |
8 |
|
incom |
⊢ ( ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ∩ 𝑆 ) = ( 𝑆 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) |
9 |
|
incom |
⊢ ( 𝑆 ∩ 𝑇 ) = ( 𝑇 ∩ 𝑆 ) |
10 |
9 6
|
eqtr3id |
⊢ ( 𝜑 → ( 𝑇 ∩ 𝑆 ) = ∅ ) |
11 |
1 2 3 4 5
|
restcls2lem |
⊢ ( 𝜑 → 𝑆 ⊆ 𝑌 ) |
12 |
1 2 3 4 7 10 11
|
restclsseplem |
⊢ ( 𝜑 → ( ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ∩ 𝑆 ) = ∅ ) |
13 |
8 12
|
eqtr3id |
⊢ ( 𝜑 → ( 𝑆 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) = ∅ ) |
14 |
1 2 3 4 7
|
restcls2lem |
⊢ ( 𝜑 → 𝑇 ⊆ 𝑌 ) |
15 |
1 2 3 4 5 6 14
|
restclsseplem |
⊢ ( 𝜑 → ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑇 ) = ∅ ) |
16 |
13 15
|
jca |
⊢ ( 𝜑 → ( ( 𝑆 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑇 ) = ∅ ) ) |