Metamath Proof Explorer

Theorem nnltp1ne

Description: Positive integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023)

Ref Expression
Assertion nnltp1ne 
|- ( ( A e. NN /\ B e. NN ) -> ( ( A + 1 ) < B <-> ( A < B /\ B =/= ( A + 1 ) ) ) )

Proof

Step Hyp Ref Expression
1 nnz 
 |-  ( A e. NN -> A e. ZZ )
2 nnz 
 |-  ( B e. NN -> B e. ZZ )
3 zltp1ne 
 |-  ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A + 1 ) < B <-> ( A < B /\ B =/= ( A + 1 ) ) ) )
4 1 2 3 syl2an 
 |-  ( ( A e. NN /\ B e. NN ) -> ( ( A + 1 ) < B <-> ( A < B /\ B =/= ( A + 1 ) ) ) )