Singleton of Element is Subset/Proof 2

Theorem

Let $S$ be a set.

Let $\set x$ be the singleton of $x$.

Then:

$x \in S \iff \set x \subseteq S$

Proof

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:

$\set x \subseteq S$

$\Box$

Sufficient Condition

Let $\set x \subseteq S$.

From the definition of a subset:

$x \in \set x \implies x \in S$

$\blacksquare$