Axiom of union

The union over the elements of a set exists.

The axiom of union states that for any set of sets F, there is a set A containing every element that is a member of some member of F:

$$\forall \mathcal{F} \, \exists A \, \forall Y \, \forall x (x \in Y \land Y \in \mathcal{F} \Rightarrow x\in A )$$

Although this formula doesn't directly assert the existence of F, the set F ⋃ F can be constructed from A (above) using the axiom schema of specification:

$\cup \mathcal {F} := \{x \in A : \exists Y (x \in Y \land Y \in \mathcal {F} \}$