For each set $X$ there is a set containing all elements of the elements of $X$, formally
$$\forall X~\exists Y~\forall xz (x\in X\wedge z\in x\Rightarrow z\in Y).$$
The set, existence of which is ensured by the axiom of union, is denoted by $\bigcup X$ or, more detailed, $\bigcup_{u\in X} u.$ Both notations mean the following set:
$$z\in \bigcup X:\Leftrightarrow z\in u\text{ for an }u\in X.$$
Corollaries: 1
Axioms: 1
