Metamath Proof Explorer

Theorem nnltlem1

Description: Positive integer ordering relation. (Contributed by NM, 21-Jun-2005)

Ref Expression
Assertion nnltlem1 
|- ( ( M e. NN /\ N e. NN ) -> ( M < N <-> M <_ ( N - 1 ) ) )

Proof

Step Hyp Ref Expression
1 nnz 
 |-  ( M e. NN -> M e. ZZ )
2 nnz 
 |-  ( N e. NN -> N e. ZZ )
3 zltlem1 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> M <_ ( N - 1 ) ) )
4 1 2 3 syl2an 
 |-  ( ( M e. NN /\ N e. NN ) -> ( M < N <-> M <_ ( N - 1 ) ) )