I need help in my proof trichotomy law of addition in $\mathbb{N}$ (Peano Axioms). I have already proved that addition is associative and commutative. Also I proved the cancellation law and some useful lemmas. Now I'm having trouble proving the following proposition:
Let $m,n \in \mathbb{N}$. Then, exactly one of the following statements is true:
- $m=n$
- There is a natural number $p \neq 0$ such that $ m = n + p$.
- There is a natural number $q \neq 0 $ such that $n = m + q$.
My try
First, I proved that two of this statements can not occur at the same time.
If $1), 2)$ are true, then $m=m+p$ and by cancellation law, $p=0$, contradiction. This is analogous for $1),3)$. Then, assume $2),3)$. Then, $m = m + q + p$, and by cancellation law, $ 0 = q + p \implies q=p=0$, a contradiction (I proved this last statement previously). Then, no more than 1 statement can be true.
Now, I need to prove that at least $1$ of the statements is true to finish the proof, but I don't know how to proceed. I know that this is a basic/classic question but I did not found any post about this in MSE. If such post exist, please let me know and sorry for reposting.
Any hints are appreciated.