Intersection of Class Exists and is Unique

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Theorem

Let $V$ be a basic universe.

Let $A \subseteq V$ be a class.

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

Then $\ds \bigcap A$ 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