The proposition is true for any set, and this proof is valid for any set, whether it be uncountable, infinite or whatever else you can think of. The crucial fact is that

$\displaystyle X=\{s\in S | s\notin f(s)\}$

is a valid subset of $\displaystyle S$ for *any* set $\displaystyle S.$ A good question to ask youself is "why wouldn't this be true for an uncountabe set?" You should eventually be convinced that there is no reason.

N.b. It is the stipulation that $\displaystyle s \in S $ in the definition of $\displaystyle X$ which rescues us from paradoxes such as Russel's Paradox.