Metamath Proof Explorer

Theorem nn0ltlem1

Description: Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004) (Proof shortened by Mario Carneiro, 16-May-2014)

Ref Expression
Assertion nn0ltlem1 
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M < N <-> M <_ ( N - 1 ) ) )

Proof

Step Hyp Ref Expression
1 nn0z 
 |-  ( M e. NN0 -> M e. ZZ )
2 nn0z 
 |-  ( N e. NN0 -> N e. ZZ )
3 zltlem1 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> M <_ ( N - 1 ) ) )
4 1 2 3 syl2an 
 |-  ( ( M e. NN0 /\ N e. NN0 ) -> ( M < N <-> M <_ ( N - 1 ) ) )