Union 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:
 * $\displaystyle \bigcup T = S$

where $\displaystyle \bigcup T$ denotes the union of $T$.

Union of $T$ Subset $S$
Let $\displaystyle x \in \bigcup T$.

By definition of union:
 * $\exists A \in T: x \in A$

By definition of $T$:
 * $\exists y \in S: A = \left\{{y}\right\}$

Then by definition of singleton:
 * $x = y$

Thus $x \in S$.

$S$ Subset Union of $T$
Let $x \in S$.

By definition of $T$:
 * $\left\{{x}\right\} \in T$

By Set is Subset of Union/Set of Sets
 * $\displaystyle \left\{{x}\right\} \subseteq \bigcup T$

By definition of singleton:
 * $x \in \left\{{x}\right\}$

Thus by definition of subset:
 * $\displaystyle x \in \bigcup T$

Thus by definition of set equality
 * $\displaystyle \bigcup T = S$