Metamath Proof Explorer

Theorem nnleltp1

Description: Positive integer ordering relation. (Contributed by NM, 13-Aug-2001) (Proof shortened by Mario Carneiro, 16-May-2014)

Ref Expression
Assertion nnleltp1 
|- ( ( A e. NN /\ B e. NN ) -> ( A <_ B <-> A < ( B + 1 ) ) )

Proof

Step Hyp Ref Expression
1 nnz 
 |-  ( A e. NN -> A e. ZZ )
2 nnz 
 |-  ( B e. NN -> B e. ZZ )
3 zleltp1 
 |-  ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B <-> A < ( B + 1 ) ) )
4 1 2 3 syl2an 
 |-  ( ( A e. NN /\ B e. NN ) -> ( A <_ B <-> A < ( B + 1 ) ) )