Mice with Woodin cardinals from a Reinhardt

Abstract

Suppose there is a Reinhardt cardinal. Then (1) $M_n(X)$ exists and is fully iterable (above $X$) for every transitive set $X$ and every $n<\omega$ (here $M_n(X)$ denotes the canonical minimal proper class inner model containing $X$ and having $n$ Woodin cardinals above the rank of $X$); and (2) Projective Determinacy holds in every set generic extension.