Metamath Proof Explorer

Theorem nngt1ne1

Description: A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004)

Ref Expression
Assertion nngt1ne1 
|- ( A e. NN -> ( 1 < A <-> A =/= 1 ) )

Proof

Step Hyp Ref Expression
1 1re 
 |-  1 e. RR
2 nnre 
 |-  ( A e. NN -> A e. RR )
3 nnge1 
 |-  ( A e. NN -> 1 <_ A )
4 leltne 
 |-  ( ( 1 e. RR /\ A e. RR /\ 1 <_ A ) -> ( 1 < A <-> A =/= 1 ) )
5 1 2 3 4 mp3an2i 
 |-  ( A e. NN -> ( 1 < A <-> A =/= 1 ) )