[Math] If $x \in\mathbb{Z}$ has the property that for all $m \in\mathbb Z$, $mx = m$, then $x = 1$

Question

I am learning proofs, and I am stuck with this proposition:

Let $x \in\mathbb{Z}$. If $x$ has the property that for all $m \in\mathbb Z$, $mx = m$, then $x = 1$.

I want to use the additive identity to get $mx = m \cdot 1$ to introduce the 1. I am tempted to simply cancel the $m$, but I am supposed to use axioms. Any idea? If $m$ would be any integer except 0, I could use the cancellation axiom. However, $m$ accounts for all integers.

Answer 1

Use $m = 1$. Then $1\cdot x = 1$ so that $x = 1$.