Intersection of Class is Subset of Intersection of Subclass

Theorem
Let $V$ be a basic universe.

Let $A$ and $B$ be classes of $V$:


 * $A \subseteq V, B \subseteq V$

such that it is not the case that $A = B = \O$.

Let $\displaystyle \bigcap A$ and $\displaystyle \bigcap B$ denote the intersection of $A$ and intersection of $B$ respectively.

Let $A$ be a subclass of $B$:
 * $A \subseteq B$

Then $\displaystyle \bigcap B$ is a subset of $\displaystyle \bigcap A$:
 * $\displaystyle \bigcap B \subseteq \displaystyle \bigcap A$

Proof
First we consider the degenerate case where $A = B = \O$.

By Intersection of Empty Class we have that:
 * $\displaystyle \bigcap \O = V$

Thus we have:
 * $\displaystyle \bigcap B = \displaystyle \bigcap A = V$

and neither $\displaystyle \bigcap B$ nor $\displaystyle \bigcap A$ are in fact sets.

So while in this case $\displaystyle \bigcap B \subseteq \displaystyle \bigcap A$, $\displaystyle \bigcap B$ is a subclass of $\displaystyle \bigcap A$ and not actually a subset.

Next we cover the case where $A = \O$ is the empty class.

It is noted that Empty Class is Subclass of All Classes, and so $A \subseteq B$.

By Intersection of Empty Class we have that:
 * $\displaystyle \bigcap A = V$

and so by definition of basic universe $\displaystyle \bigcap B \subseteq V$.

By Intersection of Non-Empty Class is Set, $\displaystyle \bigcap B$ is a set.

By definition of basic universe:
 * $\displaystyle \bigcap B \in V$

By the axiom of transitivity, $V$ is transitive.

Hence as $\displaystyle \bigcap B \in V$ it follows that:
 * $\displaystyle \bigcap B \subseteq V$

and the result holds.

By Intersection of Non-Empty Class is Set, both $\displaystyle \bigcap A$ and $\displaystyle \bigcap B$ are sets.

Let $x \in \displaystyle \bigcap B$.

Then:
 * $\forall y \in B: x \in y$

That is:
 * $y \in B \implies x \in y$

But as $A \subseteq B$, we have that:


 * $y \in A \implies y \in B$

and therefore:
 * $y \in A \implies x \in y$

That is:
 * $\forall y \in A: x \in y$

and so by definition of intersection:
 * $x \in \displaystyle \bigcap A$

Hence the result by definition of subset.