Union of Singleton

Theorem

Consider the set of sets $\mathbb S$ such that $\mathbb S$ consists of just one set $S$.

Then the union of $\mathbb S$ is $S$:

$\displaystyle \mathbb S = \set S \implies \bigcup \mathbb S = S$

Proof

Let $\mathbb S = \set S$.

Then from the definition of set union:

$\displaystyle \bigcup \mathbb S = \set {x: \exists X \in \mathbb S: x \in X}$

from which it follows directly that:

$\displaystyle \bigcup \mathbb S = \set {x: x \in S}$

as $S$ is the only set in $\mathbb S$.

That is:

$\displaystyle \bigcup \mathbb S = S$

$\blacksquare$