Is there a diagonal argument to show that if $x$ is infinite then ${\cal P}(x)$ (the power set of $x$) is smaller than $\beta x$ (the set of ultrafilters on $x$)?

(Added later. I tried commenting but it wouldn't let me!)

My reason for interest in this was roughly as follows. With AC one can prove that an infinite set $X$ has $2^{2^{|X|}}$ ultrafilters. If choice fails very badly then there might only be $|X|$-many. I was hoping that there might be a diagonal construction lurking in the background which can feed off the choice principles one supplies. As my correspondents know, i spend a lot of time thinking about Quine's NF, and one question there is: how many ultrafilters are there on the universe? If every ultrafilter is principal then very few, yes. But if there are nonprincipal ufs at all, is there some elementary argument available to show that there must be as many of them as there are sets? If there is such a diagonal construction i'd like to know. It could be useful paedogogically, too.