Prove that if m and n are integers, then either m<n or n<m, but not both.
For $\displaystyle x,y\in \mathbb{N}$ we define $\displaystyle x<y$ if and only if $\displaystyle x\in y$. What you are basically asking is to prove that $\displaystyle <$ is a linear ordering of $\displaystyle \mathbb{N}$. This should be proved by induction. Define the statement, "$\displaystyle x<n$ or $\displaystyle n<x$" for $\displaystyle n$. Assume it hold for $\displaystyle n$ and prove it holds for $\displaystyle n+1$.