Equivalence of Definitions of Synthetic Basis

Theorem
Let $S$ be a set.

1 implies 2
Let $U, V \in \BB$.

Let $x \in U \cap V$.

By hypothesis:
 * $\ds \exists \AA \subseteq \BB: U \cap V = \bigcup \AA$

By definition of union, $\exists W \in\AA : x \in W$.

By Set is Subset of Union: General Result, $W \subset U \cap V$.

Therefore:
 * $\ds \forall x \in A \cap B: \exists W \in \AA \subseteq \BB: x \in W \subseteq U \cap V$

2 implies 1
Let $U, V \in \BB$.

Define the set:
 * $\ds \AA = \set {W \in \BB: W \subseteq U \cap V} \subseteq \BB$

By Union is Smallest Superset: General Result:
 * $\ds \bigcup \AA \subseteq U \cap V$

By hypothesis:
 * $\ds \forall x \in U \cap V: \exists W \in \AA: x \in W$

Thus $\ds U \cap V \subseteq \bigcup \AA$

By definition of set equality:
 * $\ds U \cap V = \bigcup \AA$