Cardinality of Set of Singletons

Theorem

Let $S$ be a set.

Let $T = \left\{ {\left\{{x}\right\}: x \in S}\right\}$ be the set of all singletons of elements of $S$.

Then:

$\left\vert T \right\vert = \left\vert S \right\vert$

where $\left\vert S \right\vert$ denotes the cardinality of $S$.

Define a mapping $f: S \to T$:

$\forall x \in S: f \left({x}\right) = \left\{ {x}\right\}$

$\forall x, y \in S: f \left({x}\right) = f \left({y}\right) \implies x = y$

Then, by definition, $f$ is an injection.

By the definition of set $T$:

$\forall y \in T: \exists x \in S: y = f \left({x}\right)$

Then, by definition, $f$ is a surjection.

Hence, by definition, $f: S \to T$ is a bijection.

Thus, by definition, $S$ and $T$ are equivalent:

$S \sim T$

Thus by definition of cardinality:

$\left\vert T \right\vert = \left\vert S \right\vert$

$\blacksquare$