Description: A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of Kreyszig p. 30. (Contributed by Mario Carneiro, 1-May-2014)
Ref | Expression | ||
---|---|---|---|
Hypothesis | metcld.2 | |
|
Assertion | metcld2 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metcld.2 | |
|
2 | 1 | metcld | |
3 | 19.23v | |
|
4 | vex | |
|
5 | 4 | elima2 | |
6 | id | |
|
7 | elfvdm | |
|
8 | ssexg | |
|
9 | 6 7 8 | syl2anr | |
10 | nnex | |
|
11 | elmapg | |
|
12 | 9 10 11 | sylancl | |
13 | 12 | anbi1d | |
14 | 13 | exbidv | |
15 | 5 14 | bitr2id | |
16 | 15 | imbi1d | |
17 | 3 16 | bitrid | |
18 | 17 | albidv | |
19 | dfss2 | |
|
20 | 18 19 | bitr4di | |
21 | 2 20 | bitrd | |