Metamath Proof Explorer

Theorem ioossico

Description: An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017)

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

Proof

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