Metamath Proof Explorer

Theorem nn0leltp1

Description: Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004)

Ref Expression
Assertion nn0leltp1 
|- ( ( 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 zleltp1 
 |-  ( ( 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 ) ) )