Union of Singleton

Theorem
Consider the set of sets $A$ such that $A$ consists of just one set $x$:
 * $A = \set x$

Then the union of $A$ is $x$:


 * $\bigcup A = x$

Proof
Let $A = \set x$.

From the definition of set union:


 * $\bigcup \set x = \set {y: \exists z \in \set x: y \in z}$

from which it follows directly that:


 * $\bigcup \set x = \set {y: y \in x}$

as $x$ is the only set in $\set x$.

That is:


 * $\bigcup A = x$

Also see

 * Intersection of Singleton