Cardinality of Set less than Cardinality of Power Set

Theorem
Let $X$ be a set.

Then:
 * $\card X < \card {\powerset X}$

where
 * $\card X$ denotes the cardinality of $X$,
 * $\powerset X$ denotes the power set of $X$.

Proof
By No Bijection from Set to its Power Set:
 * there exist no bijections $X \to \powerset X$

Then by definition of set equivalence:
 * $X \not\sim \powerset X$

Hence by definition of cardinality:
 * $(1): \quad \card X \ne \card {\powerset X}$

By Cardinality of Set of Singletons:
 * $(2): \quad \card {\set {\set {x}: x \in X} } = \card X$

By definition of subset:
 * $\forall x \in X: \set {x} \subseteq X$

Then by definition of power set:
 * $\forall x \in X: \set {x} \in \powerset X$

Hence by definition of subset:
 * $\set {\set {x}: x \in X} \subseteq \powerset X$

Then by Subset implies Cardinal Inequality and $(2)$:
 * $\card X \le \card {\powerset X}$

Thus by $(1)$:
 * $\card X < \card {\powerset X}$