Singleton of Element is Subset/Proof 2

Theorem
Let $S$ be a set.

Let $\left\{{x}\right\}$ be the singleton of $x$.

Then:
 * $x \in S \iff \left\{{x}\right\} \subseteq S$

Necessary Condition
Let $x \in S$.

We have:
 * $\left\{{x}\right\} = \left\{{y \in S: y = x}\right\}$

From Subset of Set with Propositional Function:
 * $\left\{{x \in S: P \left({x}\right)} \right\} \subseteq S$

Hence:
 * $\left\{{x}\right\} \subseteq S$

Sufficient Condition
Let $\left\{{x}\right\} \subseteq S$.

From the definition of a subset:
 * $x \in \left\{{x}\right\} \implies x \in S$