I'm going to finish constructing real numbers from the Peano axiom.
I've just finished constructing natural numbers and integers and now I'm going to construct an inequality within the integer range, but I can't prove the most basic proposition.
How can I proove only one of these three holds?
$$\forall m,n\in\mathbb N, m>n, m=n, m<m$$
My try : It is easy to proobe that at most one of them holds. And I'll let first $m\ne n$. If $m\not>n$, then $m\notin \Bbb N+n$. I have no idea to go forth. Pleast help me.