Description: The complement of a closure is open. (Contributed by NM, 11-Sep-2006)
Ref | Expression | ||
---|---|---|---|
Hypothesis | clscld.1 | |
|
Assertion | cmclsopn | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clscld.1 | |
|
2 | 1 | clsval2 | |
3 | 2 | difeq2d | |
4 | difss | |
|
5 | 1 | ntropn | |
6 | 4 5 | mpan2 | |
7 | 1 | eltopss | |
8 | 6 7 | mpdan | |
9 | dfss4 | |
|
10 | 8 9 | sylib | |
11 | 10 6 | eqeltrd | |
12 | 11 | adantr | |
13 | 3 12 | eqeltrd | |