Axiom of Union

The Axiom of Union is a possible axiom of Material set theory: #m/def/set/zf

(\forall 𝔐 E) (\exists 𝔐 𝐵) [𝑥 \in 𝐵 ⟺ (\exists 𝑋 \in E) [𝑥 \in 𝑋]]

which is to say, for any set $E$ there exists a union $𝐵$ consisting of the elements of the elements of $E$ . It follows from the Axiom of Extensionality that such a $𝐵$ is unique, and we denote it by $⋃ E$ .

In a material set theory with classes like $N B G$ , the existence of a union class is already guaranteed by other axioms, but one requires the above axiom to guarantee that the union of sets is itself a set.

#state/tidy | #lang/en | #SemBr