Metamath Proof Explorer

Theorem eliooord

Description: Ordering implied by a member of an open interval of reals. (Contributed by NM, 17-Aug-2008) (Revised by Mario Carneiro, 9-May-2014)

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

Proof

Step Hyp Ref Expression
1 eliooxr 
 |-  ( A e. ( B (,) C ) -> ( B e. RR* /\ C e. RR* ) )
2 elioo2 
 |-  ( ( B e. RR* /\ C e. RR* ) -> ( A e. ( B (,) C ) <-> ( A e. RR /\ B < A /\ A < C ) ) )
3 1 2 syl 
 |-  ( A e. ( B (,) C ) -> ( A e. ( B (,) C ) <-> ( A e. RR /\ B < A /\ A < C ) ) )
4 3 ibi 
 |-  ( A e. ( B (,) C ) -> ( A e. RR /\ B < A /\ A < C ) )
5 3simpc 
 |-  ( ( A e. RR /\ B < A /\ A < C ) -> ( B < A /\ A < C ) )
6 4 5 syl 
 |-  ( A e. ( B (,) C ) -> ( B < A /\ A < C ) )