Does the existence of a Kuratowski-infinite set imply the existence of a Dedekind-infinite set? A set $X$ is called Dedekind-infinite if there is a injective, non-surjective map $X\to X$ and Kuratowski-infinite if $\mathcal P(X)$ is not generated by $\{\emptyset\}\cup\{\{x\}|x\in X\}$ as a sub-semilattice with respect to $\cup$.
In ZF without the axiom of infinity, any Dedekind-infinite set is also Kuratowski-infinite but the converse does not necessarily hold. However, does the existence of a Kuratowski-infinite set at least imply that a Dedekind-infinite set exists?

Yes. Easily, a set which is not Kuratowski finite is not finite. Ant the second power set of an infinite set is Dedekind infinite. 
