Consider $m\in\mathbb{N}$. Prove that if $m<n$, then $m\neq n$ and $m+1\leq n$

Question

To solve this problem I will use this result proved before:

$$m,n\in\mathbb{N} \Rightarrow m+n\neq1.$$

By definition of order, there existes $p\in\mathbb{N}$ such that $n=m+p\neq m$, using item above. Moreover, if $p=1$, we have $n=m+1$, but if $p>1$, then $n=m+p>m+1$.

Is it correct?