Intersection of Class Exists and is Unique

Theorem

Let $V$ be a basic universe.

Let $A \subseteq V$ be a class.

Let $\bigcap A$ denote the intersection of $A$.

Then $\bigcap A$ is guaranteed to exist and is unique.

Proof

By the Axiom of Specification we can create the subclass of $V$:

$\bigcap A = \set {x \in V: \forall y \in A: x \in y}$

Hence $\bigcap A$ exists.

Suppose $\QQ \subseteq V$ such that $\QQ$ and $\bigcap A$ are both the intersection of $A$.

Then:

$\QQ = \set {x \in V: \forall y \in A: x \in y}$

Thus:

$x \in \QQ \implies x \in \bigcap A$

and:

$x \in \bigcap A \implies x \in \QQ$

Hence by the Axiom of Extension:

$\QQ = \bigcap A$

and uniqueness has been demonstrated.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 5$ The union axiom