A refinement of Cantor’s theorem

It is shown that there is no surjection from the one-element subsets of a set containing an infinite co-infinite set to the infinite co-infinite subsets of that set. It is also shown that there is no surjection from the one-element subsets of an infinite set to the infinite subsets of that set. The proof can be formalized in a subtheory of both Zermelo Set Theory and New Foundations (and thus makes no use of the Axiom of Choice).