This isn't about the "meat" of your post, but: for someone who's finding Cantor's theorem hard to believe, it might be worth avoiding proof by contradiction, which can be confusing. I'd rather say "Let f be any element of (2^X)^X. Define S_f(x) = 1 - f(x)(x). Then for any x, f(x)(x) != S_f(x), which means f(x) != S_f. So for every f, there's an S_f such that there's no x that represents it."