This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lbico1

Description: The lower bound belongs to a closed-below, open-above interval. See lbicc2 . (Contributed by FL, 29-May-2014)

Ref Expression
Assertion lbico1 
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. ( A [,) B ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. RR* )
2 xrleid 
 |-  ( A e. RR* -> A <_ A )
3 2 3ad2ant1 
 |-  ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A <_ A )
4 simp3 
 |-  ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A < B )
5 elico1 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A e. ( A [,) B ) <-> ( A e. RR* /\ A <_ A /\ A < B ) ) )
6 5 3adant3 
 |-  ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( A e. ( A [,) B ) <-> ( A e. RR* /\ A <_ A /\ A < B ) ) )
7 1 3 4 6 mpbir3and 
 |-  ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. ( A [,) B ) )