You can use this to prove the Cantor-Schröder-Bernstein theorem, which asserts that whenever $A$ injects into $B$ and $B$ injects into $A$, then they are bijective. Namely, suppose that $f:A\to B$ and $g:B\to A$ are both injective functions. If there were a set $X\subset A$ such that $A-X=g[B-f[X]]$, then the function $h=(f\upharpoonright X)\cup (g^{-1}\upharpoonright A-X)$ is a bijection between $A$ and $B$. Such a set $X$ exists by the Knaster-Tarski theorem, since the powerset $P(A)$ is a complete lattice under inclusion and the function $\varphi(X)=A-g[B-f[X]]$ is $\subset$-preserving, since $$X\subset Y\implies f[X]\subset f[Y]\implies B-f[X]\supset B-f[Y]$$ $$\implies A-g[B-f[X]]\subset A-g[B-f[Y]]\implies \varphi(X)\subset\varphi(Y).$$
A fixed point $X=\varphi(x)$ means $A-X=g[B-f[X]]$.