Order Facts for Natural Numbers: Not Less Than Gives Greater or Equal, and Below n+1 Gives at Most n

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $\mathbb{N}$, the sum $n+1$ and the order $\le$ with strict order $<$ be as in The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (ii), The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (iv) and The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (v). Let $m,n\in\mathbb{N}$.

(i) If $m<n$ does not hold, then $n\le m$.

(ii) If $m<n+1$, then $m\le n$.