The Axiom of Union: $\forall x\,\exists y\,\forall z\,(z \in y \leftrightarrow \exists s\,(s \in x \wedge z \in s))$

The Axiom of Union is the sentence

$\forall x\,\exists y\,\forall z\,\bigl(z \in y \leftrightarrow \exists s\,(s \in x \wedge z \in s)\bigr)$

of the language of set theory (The first-order language of set theory: $\in$, $=$, formulas with parameters, and class abbreviations): for any set $x$ there is a set $y$ whose elements are exactly the sets $z$ belonging to some member $s$ of $x$.

The axiom removes one layer of membership: it collects the members of the members of $x$, not the members of $x$.

• The weak form is equivalent, given Separation. Some presentations assume only $\forall x\,\exists y\,\forall s\,\forall z\,((s \in x \wedge z \in s) \to z \in y)$ and then cut $y$ down with The Axiom Schema of Separation: for each formula $\varphi$, $\forall \bar p\,\forall x\,\exists y\,\forall z\,(z \in y \leftrightarrow (z \in x \wedge \varphi(z,\bar p)))$. The set obtained is the one asserted here.

• Nothing is assumed about the members of $x$. They need not be nonempty, need not be distinct, and need not be related to one another. When $x$ itself has no members the right-hand side fails for every $z$, and the axiom yields a set with no members; this is computed at $\bigcup \varnothing = \varnothing$, $\bigcup \{a\} = a$, $\bigcup \{a,b\} = a \cup b$, $\bigcap \{a\} = a$, and $\bigcap \{a,b\} = a \cap b$.