Set Union can be Derived using Axiom of Abstraction
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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:
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