Knaster-Tarski Lemma/Corollary/Power Set/Proof 2

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
Let $P = \left\{{x \in \mathcal P \left({S}\right): x \subseteq f(x) }\right\}$.

Let $\bigcup P$ be the union of $P$.

Let $x \in P$.

Then by Subset of Union, $x \subseteq \bigcup P$.

Since $f$ is increasing, $f(x) \subseteq f \left({\bigcup P}\right)$.

By the definition of $P$, $x \subseteq f \left({x}\right)$.

Thus $x \subseteq f\left({\bigcup P}\right)$ by Subset Relation is Transitive:

As this holds for all $x \in P$, $\bigcup P \subseteq f \left({\bigcup P}\right)$ by Union Smallest.

Now since $f$ is increasing, $f\left({\bigcup P}\right) \subseteq f \left({ f \left({\bigcup P}\right)}\right)$.

Thus $f \left({ \bigcup P }\right) \in P$ by the definition of $P$.

Therefore $f \left({ \bigcup P} \right) \subseteq \bigcup P$ by Subset of Union.

Since we already know that $\bigcup P \subseteq f \left({\bigcup P}\right)$, $f \left({\bigcup P}\right) = \bigcup P$ by Equality of Sets.

Thus $\bigcup P$ is a fixed point of $f$.