Theorem icossico

Description: Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Thierry Arnoux, 21-Sep-2017)

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

Proof

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