discrete mathematics – Cantors Theorem Proof

I am working on my own proof for cantors theorem that given any set A, there does not exist a function f: A -> P(A) that is onto. I was wondering if it would be possible to prove this by showing that the cardinality of A is less than P(A) using the proof that the elements of set A is n and P(A) is 2^n so n < 2^n for all natural numbers (by induction). and due to the cardinality being less is it not surjective since not all elements of the codomain are mapped by the domain?