Subset of Set with Propositional Function

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Theorem

Let $S$ be a set.

Let $P: S \to \left\{{\text{true}, \text{false}}\right\}$ be a propositional function on $S$.

Then:

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle s \in \left\{ {x \in S: P \left({x}\right)} \right\}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle s \in \left\{ {x \in S \land P \left({x}\right)}\right\}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle s \in \left\{ {x \in S} \right\}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Rule of Simplification
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle s \in S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of an element
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle \left\{ {x \in S: P \left({x}\right)}\right\} \subseteq S$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of a subset

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle s \in \left\{ {x \in S: P \left({x}\right)} \right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle s \in \left\{ {x \in S \land P \left({x}\right)}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle s \in \left\{ {x \in S} \right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Rule of Simplification
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle s \in S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of an element
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left\{ {x \in S: P \left({x}\right)}\right\} \subseteq S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of a subset