Metamath Proof Explorer

Theorem icossicc

Description: A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016)

Ref Expression
Assertion icossicc 
|- ( A [,) B ) C_ ( A [,] B )

Proof

Step Hyp Ref Expression
1 df-ico 
 |-  [,) = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a <_ x /\ x < b ) } )
2 df-icc 
 |-  [,] = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a <_ x /\ x <_ b ) } )
3 idd 
 |-  ( ( A e. RR* /\ w e. RR* ) -> ( A <_ w -> A <_ w ) )
4 xrltle 
 |-  ( ( w e. RR* /\ B e. RR* ) -> ( w < B -> w <_ B ) )
5 1 2 3 4 ixxssixx 
 |-  ( A [,) B ) C_ ( A [,] B )