Metamath Proof Explorer

Theorem posdifi

Description: Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001)

Ref Expression
Hypotheses lt2.1 
|- A e. RR
lt2.2 
|- B e. RR
Assertion posdifi 
|- ( A < B <-> 0 < ( B - A ) )

Proof

Step Hyp Ref Expression
1 lt2.1 
 |-  A e. RR
2 lt2.2 
 |-  B e. RR
3 posdif 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B - A ) ) )
4 1 2 3 mp2an 
 |-  ( A < B <-> 0 < ( B - A ) )