Metamath Proof Explorer

Theorem nnnlt1

Description: A positive integer is not less than one. (Contributed by NM, 18-Jan-2004) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion nnnlt1 
|- ( A e. NN -> -. A < 1 )

Proof

Step Hyp Ref Expression
1 nnge1 
 |-  ( A e. NN -> 1 <_ A )
2 1re 
 |-  1 e. RR
3 nnre 
 |-  ( A e. NN -> A e. RR )
4 lenlt 
 |-  ( ( 1 e. RR /\ A e. RR ) -> ( 1 <_ A <-> -. A < 1 ) )
5 2 3 4 sylancr 
 |-  ( A e. NN -> ( 1 <_ A <-> -. A < 1 ) )
6 1 5 mpbid 
 |-  ( A e. NN -> -. A < 1 )