Metamath Proof Explorer

Theorem elioo1

Description: Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006) (Revised by Mario Carneiro, 3-Nov-2013)

Ref Expression
Assertion elioo1 
|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,) B ) <-> ( C e. RR* /\ A < C /\ C < B ) ) )

Proof

Step Hyp Ref Expression
1 df-ioo 
 |-  (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
2 1 elixx1 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,) B ) <-> ( C e. RR* /\ A < C /\ C < B ) ) )