Metamath Proof Explorer

Theorem iccssioo

Description: Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015)

Ref Expression
Assertion iccssioo 
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < C /\ D < B ) ) -> ( C [,] D ) C_ ( A (,) B ) )

Proof

Step Hyp Ref Expression
1 df-ioo 
 |-  (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
2 df-icc 
 |-  [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
3 xrltletr 
 |-  ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) -> ( ( A < C /\ C <_ w ) -> A < w ) )
4 xrlelttr 
 |-  ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) -> ( ( w <_ D /\ D < B ) -> w < B ) )
5 1 2 3 4 ixxss12 
 |-  ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < C /\ D < B ) ) -> ( C [,] D ) C_ ( A (,) B ) )