Metamath Proof Explorer

Theorem nngt0

Description: A positive integer is positive. (Contributed by NM, 26-Sep-1999)

Ref Expression
Assertion nngt0 
|- ( A e. NN -> 0 < A )

Proof

Step Hyp Ref Expression
1 nnre 
 |-  ( A e. NN -> A e. RR )
2 nnge1 
 |-  ( A e. NN -> 1 <_ A )
3 0lt1 
 |-  0 < 1
4 0re 
 |-  0 e. RR
5 1re 
 |-  1 e. RR
6 ltletr 
 |-  ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 <_ A ) -> 0 < A ) )
7 4 5 6 mp3an12 
 |-  ( A e. RR -> ( ( 0 < 1 /\ 1 <_ A ) -> 0 < A ) )
8 3 7 mpani 
 |-  ( A e. RR -> ( 1 <_ A -> 0 < A ) )
9 1 2 8 sylc 
 |-  ( A e. NN -> 0 < A )