Show that a function $f:P(X)\to P(X)$ preserving the subset relation has a fixed point We have a map $f:P(X)\to P(X)$, where $P(X)$ means the part of $X$ and the function is  monotone (by considering inclusion "$\subseteq$"). So $\forall \space A\subseteq B $ we have $f(A)\subseteq f(B)$.
Show that this map has a fixed point.
This claim is used in some proofs of Cantor–Schröder–Bernstein theorem, for example, see proof 3 on ProofWiki (current revision).

HINT: Consider the set $\bigcup\{A\subseteq X:A\subseteq f(A)\}$. (Be sure to show that there is at least one $A\subseteq X$ such that $A\subseteq f(A)$.)
