Metamath Proof Explorer

Theorem lenlti

Description: 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999)

Ref Expression
Hypotheses lt.1 
|- A e. RR
lt.2 
|- B e. RR
Assertion lenlti 
|- ( A <_ B <-> -. B < A )

Proof

Step Hyp Ref Expression
1 lt.1 
 |-  A e. RR
2 lt.2 
 |-  B e. RR
3 lenlt 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
4 1 2 3 mp2an 
 |-  ( A <_ B <-> -. B < A )