Existence of Unique Subgroup Generated by Subset/Singleton Generator

Theorem
Let $\left({G, \circ}\right)$ be a group. Let $a \in G$.

Then $H = \left\langle {a}\right\rangle = \left\{{a^n: n \in \Z}\right\}$ is the unique smallest subgroup of $G$ such that $a \in H$.

That is:
 * $K \le G: a \in K \implies H \subseteq K$

Proof
From Powers of Element form Subgroup, $H = \left\{{a^n: n \in \Z}\right\}$ is a subgroup of $G$.

Let $K \le G: a \in K$.

Then $\forall n \in \Z: a^n \in K$.

Thus, $H \subseteq K$.