Metamath Proof Explorer

Theorem iooss2

Description: Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007) (Revised by Mario Carneiro, 3-Nov-2013)

		Ref	Expression
	Assertion	iooss2	\|- ( ( C e. RR* /\ B <_ C ) -> ( A (,) B ) C_ ( A (,) C ) )

Step	Hyp	Ref	Expression
1		df-ioo	\|- (,) = ( x e. RR* , y e. RR* \|-> { z e. RR* \| ( x < z /\ z < y ) } )
2		xrltletr	\|- ( ( w e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( w < B /\ B <_ C ) -> w < C ) )
3	1 1 2	ixxss2	\|- ( ( C e. RR* /\ B <_ C ) -> ( A (,) B ) C_ ( A (,) C ) )

Step

Hyp

Ref

Expression

 |-  (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )

 |-  ( ( w e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( w < B /\ B <_ C ) -> w < C ) )

1 1 2

 |-  ( ( C e. RR* /\ B <_ C ) -> ( A (,) B ) C_ ( A (,) C ) )