Description: A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007)
Ref | Expression | ||
---|---|---|---|
Hypothesis | clscld.1 | |
|
Assertion | ntrcls0 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clscld.1 | |
|
2 | simpl | |
|
3 | 1 | clsss3 | |
4 | 1 | sscls | |
5 | 1 | ntrss | |
6 | 2 3 4 5 | syl3anc | |
7 | 6 | 3adant3 | |
8 | sseq2 | |
|
9 | 8 | 3ad2ant3 | |
10 | 7 9 | mpbid | |
11 | ss0 | |
|
12 | 10 11 | syl | |