Metamath Proof Explorer

Theorem iooinlbub

Description: An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion iooinlbub 
|- ( ( A (,) B ) i^i { A , B } ) = (/)

Proof

Step Hyp Ref Expression
1 disjr 
 |-  ( ( ( A (,) B ) i^i { A , B } ) = (/) <-> A. x e. { A , B } -. x e. ( A (,) B ) )
2 elpri 
 |-  ( x e. { A , B } -> ( x = A \/ x = B ) )
3 lbioo 
 |-  -. A e. ( A (,) B )
4 eleq1 
 |-  ( x = A -> ( x e. ( A (,) B ) <-> A e. ( A (,) B ) ) )
5 3 4 mtbiri 
 |-  ( x = A -> -. x e. ( A (,) B ) )
6 ubioo 
 |-  -. B e. ( A (,) B )
7 eleq1 
 |-  ( x = B -> ( x e. ( A (,) B ) <-> B e. ( A (,) B ) ) )
8 6 7 mtbiri 
 |-  ( x = B -> -. x e. ( A (,) B ) )
9 5 8 jaoi 
 |-  ( ( x = A \/ x = B ) -> -. x e. ( A (,) B ) )
10 2 9 syl 
 |-  ( x e. { A , B } -> -. x e. ( A (,) B ) )
11 1 10 mprgbir 
 |-  ( ( A (,) B ) i^i { A , B } ) = (/)