Q:

How to prove that the union of sets is a set?

Consider a collection of sets $S_{i}$ with $i \in I$ and suppose that the index set $I$ is also a set. How can we prove that the union $\bigcup_{i \in I}S_{i}$ is also a set?

A:

The axiom schema of replacement assures that $\{S_i\mid i\in I\}$ is a set.
According to the axiom of sum for every set $A$ there is a set $\bigcup A$ such that
$$x\in\bigcup A\iff x\in a\text{ for some }a\in A$$
Applying this on the set $\{S_i\mid i\in I\}$ we find that $\bigcup_{i\in I}S_i$ is a set.

