Union of Set of Dense-in-itself Sets is Dense-in-itself

Theorem
Let $T$ be a topological space.

Let $\FF \subseteq \powerset T$ such that
 * every element of $\FF$ is dense-in-itself.

Then the union $\bigcup \FF$ is also dense-in-itself.

Proof
By Dense-in-itself iff Subset of Derivative:
 * $\forall A \in \FF: A \subseteq A'$

where $A'$ denotes the derivative of $A$.

Then by Set Union Preserves Subsets:
 * $\ds \bigcup \FF \subseteq \bigcup_{A \mathop \in \FF} A'$

By Union of Derivatives is Subset of Derivative of Union:
 * $\ds \bigcup_{A \mathop \in \FF} A' \subseteq \paren {\bigcup \FF}'$

Then by Subset Relation is Transitive:
 * $\ds \bigcup \FF \subseteq \paren {\bigcup \FF}'$

The result follows by Dense-in-itself iff Subset of Derivative.