Power Set of Natural Numbers has Cardinality of Continuum

Theorem
Let $\N$ denote the set of natural numbers.

Let $\powerset \N$ denote the power set of $\N$.

Let $\card {\powerset \N}$ denote the cardinality of $\powerset \N$.

Let $\mathfrak c = \card \R$ denote the cardinality of the continuum.

Then:
 * $\mathfrak c = \card {\powerset \N}$