Subset of Set with Propositional Function

Theorem
$$\left\{{x \in S: P \left({x}\right)} \right\} \subseteq S$$

In this context, $$P \left({x}\right)$$ is a propositional function which returns a value of true or false depending both on the value of $$x$$ and the nature of $$P$$.

Proof
$$a \in \left\{{x \in S: P \left({x}\right)} \right\}$$

$$\Longrightarrow a \in \left\{{x \in S \land P \left({x}\right)}\right\}$$

$$\Longrightarrow a \in \left\{{x \in S} \right\}$$ Rule of Simplification

$$\Longrightarrow a \in S$$

$$\Longrightarrow \left\{{x \in S: P \left({x}\right)}\right\} \subseteq S$$ Subset definition