Theorem iccss

Description: Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009) (Revised by Mario Carneiro, 20-Feb-2015)

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

Proof

Step	Hyp	Ref	Expression
1		rexr	|- ( A e. RR -> A e. RR* )
2		rexr	|- ( B e. RR -> B e. RR* )
3	1 2	anim12i	|- ( ( A e. RR /\ B e. RR ) -> ( A e. RR* /\ B e. RR* ) )
4		df-icc	|- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
5		xrletr	|- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) -> ( ( A <_ C /\ C <_ w ) -> A <_ w ) )
6		xrletr	|- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) -> ( ( w <_ D /\ D <_ B ) -> w <_ B ) )
7	4 4 5 6	ixxss12	|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )
8	3 7	sylan	|- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )