I have just been learning about Cantor's Theorem, which has been stated in by book as "the carndinality of every set is strictly less than the cardinality of it's power set", and I have a question about the theorem.
In the proof I have been given for Cantor's Theorem, the argument is put forward that the power set contains a singleton set corresponding to each element of the original set, and hence cardX $\le$ cardP(X). They then must just prove that X$\ne$P(X), that is that there exists no surjective function from X to P(X), and hence there can exist no bijective function between them, so the theorem must be true.
Is this introductory step correctly stated? If it is, then I don't understand why Cantor's Theorem doesn't trivially hold true. Using the same reasoning, could it not be said that if the power set contains a singleton set corresponding to each element in the original set, but it also contains the empty set, then surely the power set must have strictly greater cardinality than the original set?
Does this reasoning perhaps not apply when dealing with infinite sets? If not, please try and provide some intuitive reason why the argument I have supplied above fails.