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 $(1)$:

$\card X < \card {\powerset X}$

$\blacksquare$

Sources

• Mizar article CARD_1:14