The union $\bigcup x$ of a set, and the binary union $a \cup b := \bigcup \{a,b\}$

Let $x$ be a set. The Axiom of Union: $\forall x\,\exists y\,\forall z\,(z \in y \leftrightarrow \exists s\,(s \in x \wedge z \in s))$ gives a set whose elements are exactly the sets $z$ belonging to some member of $x$, and The Axiom of Extensionality: $\forall x\,\forall y\,(\forall z\,(z \in x \leftrightarrow z \in y) \to x = y)$ shows there is only one such set; it is written $\bigcup x$. Thus $\bigcup x$ is the set whose elements are exactly the elements of the elements of $x$, and $a \cup b := \bigcup\{a,b\}$ is the binary union of $a$ and $b$, formed from the unordered pair of The unordered pair $\{x,y\}$ and the singleton $\{x\} = \{x,x\}$:

$z \in \bigcup x \leftrightarrow \exists s\,(s \in x \wedge z \in s).$

The membership criterion for the binary union, that $z \in a \cup b$ holds exactly when $z \in a$ or $z \in b$, is proved at $\bigcup \varnothing = \varnothing$, $\bigcup \{a\} = a$, $\bigcup \{a,b\} = a \cup b$, $\bigcap \{a\} = a$, and $\bigcap \{a,b\} = a \cap b$ rather than assumed here.

• One layer only. $\bigcup x$ collects the elements of the elements of $x$, not the elements of $x$. For a set of sets of sets the operation therefore has to be applied twice, and that iteration is what makes $\bigcup\bigcup R$ the right ambient set for the domain and range of a relation.

• Notation. $\bigcup$ with a single set argument is the primitive operation supplied by the axiom; $\cup$ between two sets is the derived binary one. The indexed form $\bigcup_{i \in I} A_i$ is a further abbreviation, defined once indexed families are available.