Metamath Proof Explorer


Theorem iooid

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 ) = (/)

Proof

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 ) = (/)