Description: An open interval with identical lower and upper bounds is empty. (Contributed by NM, 21-Jun-2007) (Revised by Mario Carneiro, 3-Nov-2013)
Ref | Expression | ||
---|---|---|---|
Assertion | iooid | |- ( A (,) A ) = (/) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrleid | |- ( A e. RR* -> A <_ A ) |
|
2 | 1 | adantr | |- ( ( A e. RR* /\ A e. RR* ) -> A <_ A ) |
3 | ioo0 | |- ( ( A e. RR* /\ A e. RR* ) -> ( ( A (,) A ) = (/) <-> A <_ A ) ) |
|
4 | 2 3 | mpbird | |- ( ( A e. RR* /\ A e. RR* ) -> ( A (,) A ) = (/) ) |
5 | ndmioo | |- ( -. ( A e. RR* /\ A e. RR* ) -> ( A (,) A ) = (/) ) |
|
6 | 4 5 | pm2.61i | |- ( A (,) A ) = (/) |