Knaster-Tarski Lemma/Corollary/Power Set/Proof 1

Theorem
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $f: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$ be a $\subseteq$-increasing mapping.

That is, suppose that for all $T, U \in \mathcal P \left({S}\right)$:


 * $T \subseteq U \implies f \left({T}\right) \subseteq f\left({U}\right)$

Then $f$ has a fixed point.

Proof
By the Knaster-Tarski Lemma: Power Set, $f$ has a least fixed point.

Thus it has a fixed point.