Set Union can be Derived using Axiom of Abstraction

Theorem

Let $a$ be a set of sets.

By application of the Axiom of Abstraction, the union $\bigcup a$ can be formed.

Hence the union $\bigcup a$ can be derived as a valid object in Frege set theory.

Proof

Let $P$ be the property defined as:

$\forall x: \map P x := \paren {\exists y: y \in a \land x \in y}$

where $\land$ is the conjunction operator.

That is, $\map P x$ if and only if:

$x$ is an element of some set $y$, where $y$ is one of the sets that comprise the elements of $a$.

Hence, using the Axiom of Abstraction, we form the set:

$\bigcup a := \set {x: \exists y: y \in a \land x \in y}$

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 7$ Frege set theory