Cardinality of Set Union/Corollary

Theorem
Let $\mathcal S$ be an algebra of sets.

Let $S_1, S_2, \ldots, S_n$ be finite sets of $\mathcal S$ which are pairwise disjoint.

Then:
 * $\displaystyle \left\vert{\bigcup_{i \mathop = 1}^n S_i}\right\vert = \sum_{i \mathop = 1}^n \left\vert{S_i}\right\vert$

Proof
As $S_1, S_2, \ldots, S_n$ are pairwise disjoint, their intersections are all empty.

The Cardinality of Set Union holds, but from Cardinality of Empty Set, all the terms apart from the first vanish.