Metamath Proof Explorer

Theorem ltnle

Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005)

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

Proof

Step Hyp Ref Expression
1 lenlt 
 |-  ( ( B e. RR /\ A e. RR ) -> ( B <_ A <-> -. A < B ) )
2 1 ancoms 
 |-  ( ( A e. RR /\ B e. RR ) -> ( B <_ A <-> -. A < B ) )
3 2 con2bid 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -. B <_ A ) )