Singleton of Element is Subset/Proof 2

Necessary Condition
Let $x \in S$.

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

From Subset of Set with Propositional Function:
 * $\set {x \in S: \map P x} \subseteq S$

Hence:
 * $\set x \subseteq S$

Sufficient Condition
Let $\set x \subseteq S$.

From the definition of a subset:
 * $x \in \set x \implies x \in S$