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

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 $$ \{ a_1, \ldots, a_n \} = \{ a_1 \} \cup \cdots \cup \{ a_n \} $$