Metamath Proof Explorer

Theorem nn0lem1lt

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

Ref Expression
Assertion nn0lem1lt 
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> ( M - 1 ) < N ) )

Proof

Step Hyp Ref Expression
1 nn0z 
 |-  ( M e. NN0 -> M e. ZZ )
2 nn0z 
 |-  ( N e. NN0 -> N e. ZZ )
3 zlem1lt 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M - 1 ) < N ) )
4 1 2 3 syl2an 
 |-  ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> ( M - 1 ) < N ) )