Metamath Proof Explorer

Theorem nn0nlt0

Description: A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion nn0nlt0 
|- ( A e. NN0 -> -. A < 0 )

Proof

Step Hyp Ref Expression
1 nn0ge0 
 |-  ( A e. NN0 -> 0 <_ A )
2 0re 
 |-  0 e. RR
3 nn0re 
 |-  ( A e. NN0 -> A e. RR )
4 lenlt 
 |-  ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> -. A < 0 ) )
5 2 3 4 sylancr 
 |-  ( A e. NN0 -> ( 0 <_ A <-> -. A < 0 ) )
6 1 5 mpbid 
 |-  ( A e. NN0 -> -. A < 0 )