Reduced Residue System is Subset of Set of All Residue Classes

Theorem
Let $$\mathbb{Z}_m$$ be the set of integers modulo $m$.

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

Then $$\forall m \in \mathbb{Z}: m > 1: \varnothing \subset \mathbb{Z}'_m \subset \mathbb{Z}_m$$.

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

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


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

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

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

Thus $$\mathbb{Z}'_m \subset \mathbb{Z}_m$$.


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

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

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