  We introduce "$n$-choiceless" supercompact and extendible cardinals in
Zermelo-Fraenkel set theory without the Axiom of Choice. We prove relations
between these cardinals and Vop\v{e}nka's Principle similar to those of
Bagaria's work in his papers "$C^{(n)}$-Cardinals" and "More on the
Preservation of Large Cardinals Under Class Forcing." We use these relations to
characterize Berkeley cardinals in terms of a restricted form of Vop\v{e}nka's
Principle. Finally, we establish the equiconsistency of the "$n$-choiceless"
extendible cardinals with their original counterparts, and study the
consistency strength of other relevant theories.
