Metamath Proof Explorer

Theorem lenlt

Description: 'Less than or equal to' expressed in terms of 'less than'. (Contributed by NM, 13-May-1999)

Ref Expression
Assertion lenlt 
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )

Proof

Step Hyp Ref Expression
1 rexr 
 |-  ( A e. RR -> A e. RR* )
2 rexr 
 |-  ( B e. RR -> B e. RR* )
3 xrlenlt 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
4 1 2 3 syl2an 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )