Description: { (/) } is the only topology with one element. (Contributed by FL, 18-Aug-2008)
Ref | Expression | ||
---|---|---|---|
Assertion | en1top | |- ( J e. Top -> ( J ~~ 1o <-> J = { (/) } ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0opn | |- ( J e. Top -> (/) e. J ) |
|
2 | en1eqsn | |- ( ( (/) e. J /\ J ~~ 1o ) -> J = { (/) } ) |
|
3 | 2 | ex | |- ( (/) e. J -> ( J ~~ 1o -> J = { (/) } ) ) |
4 | 1 3 | syl | |- ( J e. Top -> ( J ~~ 1o -> J = { (/) } ) ) |
5 | id | |- ( J = { (/) } -> J = { (/) } ) |
|
6 | 0ex | |- (/) e. _V |
|
7 | 6 | ensn1 | |- { (/) } ~~ 1o |
8 | 5 7 | eqbrtrdi | |- ( J = { (/) } -> J ~~ 1o ) |
9 | 4 8 | impbid1 | |- ( J e. Top -> ( J ~~ 1o <-> J = { (/) } ) ) |