Theorem iooin

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

	Ref	Expression
Assertion	iooin	|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = ( if ( A <_ C , C , A ) (,) if ( B <_ D , B , D ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-ioo	|- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
2		xrmaxlt	|- ( ( A e. RR* /\ C e. RR* /\ z e. RR* ) -> ( if ( A <_ C , C , A ) < z <-> ( A < z /\ C < z ) ) )
3		xrltmin	|- ( ( z e. RR* /\ B e. RR* /\ D e. RR* ) -> ( z < if ( B <_ D , B , D ) <-> ( z < B /\ z < D ) ) )
4	1 2 3	ixxin	|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = ( if ( A <_ C , C , A ) (,) if ( B <_ D , B , D ) ) )