Proof of Cantor's Theorem

一邊爬象山一邊想，為什麼 P(N) ~ R？因為 card(N) < card(R)，應先確定 card(N) < card(P(N)) ，所以我寫出以下證明。
印象中曾在 Wikipedia 瀏覽過 Cantor's Theorem（https://goo.gl/6XoGTZ），現在自己想通了 ！
爬山不忘想數學和賞鳥。

Cantor's Theorem: Given any set A, there does not exist a function f : A → P(A) that is onto.

Assume that f is an onto function that maps A to the power set of A. Then there is a subset X of A containing all elements that are not contained in their images. X={x|x∉f(x)} X is not empty because the pre-image of the empty set must be in X. X is an element of the power set. Let x be the pre-image of X. X is the image of x. If x∈X, then f(x) does not contain x. Therefore, x∉X. If  x∉X, then x∈X by the definition of X. (A contradiction.) Therefore, X does not have a pre-image. Therefore, f cannot be onto.