Singleton of Power Set less Empty Set is Minimal Subset

Theorem
Let $S$ be a set which is non-empty.

Let $\mathcal C = \mathcal P \left({S}\right) \setminus \varnothing$, that is, the power set of $S$ without the empty set.

Let $x \in S$.

Then $\left\{{x}\right\}$ is a minimal element of the ordered structure $\left({\mathcal C, \subseteq}\right)$.

Proof
Let $y \in \mathcal C$ such that $y \subseteq \left\{{x}\right\}$.

We have that $\varnothing \notin \mathcal C$.

Therefore:
 * $\exists z \in S: z \in y$

But as $y \subseteq \left\{{x}\right\}$ it follows that:
 * $z \in \left\{{x}\right\}$

and so by definition of singleton:
 * $z = x$

and so:
 * $y = \left\{{x}\right\}$

and so:
 * $y = x$

Thus, by definition, $\left\{{x}\right\}$ is a minimal element of $\left({\mathcal C, \subseteq}\right)$.