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.