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Axiom of union

A set is an arbitrary collection of objects. The axiom of union allows us to combine the objects in many different sets and make them members of a single new set. It says we can go down two levels taking not the members of a set but the members of members of a set and combine them into a new set.

$$\forall x \exists y \hspace{.05in} \forall z \hspace{.1in} z \in y \equiv (\exists t \hspace{.05in} z \in t \wedge t \in x)$$

This says for every set $x$ there exists a set $y$ that is the union of all the members of $x$. For every $z$ that belongs to $y$ there must be some set $t$ such that $t$ belongs to $x$ and $z$ belongs to $t$.