Singleton of Element is Subset

Theorem

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

Proof
First suppose that $x \in S$.

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

Thus from $\left\{{x \in S: P \left({x}\right)} \right\} \subseteq S$ as proved here, $\left\{{x}\right\} \subseteq S$.

Now suppose $\left\{{x}\right\} \subseteq S$.

Then $x \in \left\{{x}\right\} \implies x \in S$ from the definition of a subset.