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: $\mathrm{\forall }x\mathrm{\exists }y\mathrm{\forall }z(z\in y\leftrightarrow \mathrm{\exists }s(s\in x\wedge z\in s))$ gives a set whose elements are exactly the sets $z$ belonging to some member of $x$, and The Axiom of Extensionality: $\mathrm{\forall }x\mathrm{\forall }y(\mathrm{\forall }z(z\in x\leftrightarrow z\in y)\to x=y)$ 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 $\{x\}=\{x,x\}$:

$z\in \bigcup x\leftrightarrow \mathrm{\exists }s(s\in x\wedge z\in s)\mathrm{.}$

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 $\bigcup \mathrm{\varnothing }=\mathrm{\varnothing }$, $\bigcup \{a\}=a$, $\bigcup \{a,b\}=a\cup b$, $\bigcap \{a\}=a$, 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.