Theorem elixx1

Description: Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013)

	Ref	Expression
Hypothesis	ixx.1	|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
Assertion	elixx1	|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> ( C e. RR* /\ A R C /\ C S B ) ) )

Proof

Step	Hyp	Ref	Expression
1		ixx.1	|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
2	1	ixxval	|- ( ( A e. RR* /\ B e. RR* ) -> ( A O B ) = { z e. RR* | ( A R z /\ z S B ) } )
3	2	eleq2d	|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> C e. { z e. RR* | ( A R z /\ z S B ) } ) )
4		breq2	|- ( z = C -> ( A R z <-> A R C ) )
5		breq1	|- ( z = C -> ( z S B <-> C S B ) )
6	4 5	anbi12d	|- ( z = C -> ( ( A R z /\ z S B ) <-> ( A R C /\ C S B ) ) )
7	6	elrab	|- ( C e. { z e. RR* | ( A R z /\ z S B ) } <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
8		3anass	|- ( ( C e. RR* /\ A R C /\ C S B ) <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
9	7 8	bitr4i	|- ( C e. { z e. RR* | ( A R z /\ z S B ) } <-> ( C e. RR* /\ A R C /\ C S B ) )
10	3 9	bitrdi	|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> ( C e. RR* /\ A R C /\ C S B ) ) )