Reduced Residue System is Subset of Set of All Residue Classes

Theorem
Let $$\Z_m$$ be the set of integers modulo $m$.

Let $$\Z'_m$$ be the set of integers coprime to $$m$$ in $$\Z_m$$.

Then $$\forall m \in \Z: m > 1: \varnothing \subset \Z'_m \subset \Z_m$$.

Proof
By its definition, we have $$\Z'_m = \left\{{x \in \Z_m: x \perp m}\right\}$$.

From Subset of Set with Propositional Function, we have $$\Z'_m \subseteq \Z_m$$.


 * As $$\gcd \left\{{m, 0}\right\} = m$$ it follows that $$m > 1 \implies \gcd \left\{{m, 0}\right\} \ne 1$$.

So $$\left[\!\left[{0}\right]\!\right]_m \notin \Z'_m$$.

However, $$\left[\!\left[{0}\right]\!\right]_m \in \Z_m$$, so $$\Z'_m \ne \Z_m$$.

Thus $$\Z'_m \subset \Z_m$$.


 * Now, note that $$\left[\!\left[{1}\right]\!\right]_m = 1 \implies 1 \perp m$$.

So $$\forall m \in \Z: \left[\!\left[{1}\right]\!\right]_m \in \Z'_m$$.

Thus $$\Z'_m \ne \varnothing$$ and therefore $$\varnothing \subset \Z'_m$$.