Cantor's Theorem (Strong Version)

Theorem
Let $S$ be a set.

Let $\mathcal P^n \left({S}\right)$ be defined recursively by:
 * $\mathcal P^n \left({S}\right) = \begin{cases}

S & : n = 0 \\ \mathcal P \left({\mathcal P^{n-1} \left({S}\right)}\right) & : n > 0 \end{cases}$ where $\mathcal P \left({S}\right)$ denotes the power set of $S$.

Then $S$ is not equivalent to $\mathcal P^n \left({S}\right)$ for any $n > 0$.

Also see

 * Cantor's Theorem