Metamath Proof Explorer

Theorem elioore

Description: A member of an open interval of reals is a real. (Contributed by NM, 17-Aug-2008) (Revised by Mario Carneiro, 3-Nov-2013)

Ref Expression
Assertion elioore 
|- ( A e. ( B (,) C ) -> A e. RR )

Proof

Step Hyp Ref Expression
1 elioo3g 
 |-  ( A e. ( B (,) C ) <-> ( ( B e. RR* /\ C e. RR* /\ A e. RR* ) /\ ( B < A /\ A < C ) ) )
2 3ancomb 
 |-  ( ( B e. RR* /\ C e. RR* /\ A e. RR* ) <-> ( B e. RR* /\ A e. RR* /\ C e. RR* ) )
3 xrre2 
 |-  ( ( ( B e. RR* /\ A e. RR* /\ C e. RR* ) /\ ( B < A /\ A < C ) ) -> A e. RR )
4 2 3 sylanb 
 |-  ( ( ( B e. RR* /\ C e. RR* /\ A e. RR* ) /\ ( B < A /\ A < C ) ) -> A e. RR )
5 1 4 sylbi 
 |-  ( A e. ( B (,) C ) -> A e. RR )