axiom of union

For any $X$ there exists a set $Y=\bigcup X$.

The Axiom of Union is an axiom of Zermelo-Fraenkel set theory. In symbols, it reads

$$\forall X\exists Y\forall u(u\in Y\leftrightarrow \exists z(z\in X\wedge u\in z)).$$

Notice that this means that $Y$ is the set of elements of all elements of $X$. More succinctly, the union of any set of sets is a set. By Extensionality, the set $Y$ is unique. $Y$ is called the union of $X$.

In particular, the Axiom of Union, along with the Axiom of Pairing allows us to define

$$X\cup Y=\bigcup \{X,Y\},$$

as well as the triple

$$\{a,b,c\}=\{a,b\}\cup \{c\}$$

and therefore the $n$-tuple

$$\{a_1,\mathrm{\dots },a_n\}=\{a_1\}\cup \mathrm{\cdots }\cup \{a_n\}$$