Aleph Zero is less than Cardinality of Continuum

Theorem
$\aleph_0 < \mathfrak c$

where
 * $\aleph$ denotes the aleph mapping,
 * $\mathfrak c$ denotes continuum, the cardinality of real numbers.

Proof
By Power Set of Natural Numbers is Cardinality of Continuum:
 * $\mathfrak c = \card {\powerset \N}$

where:
 * $\powerset \N$ denotes the power set of $\N$
 * $\card {\powerset \N}$ denotes the cardinality of $\powerset \N$.

By Cardinality of Set less than Cardinality of Power Set:
 * $\card \N < \card {\powerset \N}$

Thus by Aleph Zero equals Cardinality of Naturals:
 * $\aleph_0 < \mathfrak c$