Cantor's Theorem (Strong Version)

Theorem
Let $S$ be a set.

Let $\map {\PP^n} S$ be defined recursively by:
 * $\map {\PP^n} S = \begin{cases}

S & : n = 0 \\ \powerset {\map {\PP^{n - 1} } S} & : n > 0 \end{cases}$ where $\powerset S$ denotes the power set of $S$.

Then $S$ is not equivalent to $\map {\PP^n} S$ for any $n > 0$.

Also see

 * Cantor's Theorem