Class Union Exists and is Unique

Theorem
Let $V$ be a basic universe.

Let $A \subseteq V$ be a class.

Let $\ds \bigcup A$ denote the union of $A$.

Then $\ds \bigcup A$ is guaranteed to exist and is unique.

Proof
By the Axiom of Specification the union of $A$ can be created:


 * $\ds \bigcup A := \set {x: \exists y: x \in y \land y \in A}$

Hence $\ds \bigcup A$ exists.

Let $B$ and $C$ both be unions of $A$.

From the definition of union:
 * $\forall A$:
 * $x \in B \iff \exists y \in A: x \in y$
 * $x \in C \iff \exists y \in A: x \in y$

From Biconditional is Commutative and Biconditional is Transitive:
 * $x \in B \iff x \in C$

By the Axiom of Extension:
 * $B = C$

Hence the union of $A$ is unique.