Equivalence of Definitions of Synthetic Basis

Theorem
Let $S$ be a set.

1 implies 2
Let $U, V \in \mathcal B$.

Let $x \in U \cap V$.

By hypothesis:
 * $\displaystyle \exists \mathcal A \subseteq \mathcal B: U \cap V = \bigcup \mathcal A$

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

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

Therefore:
 * $\displaystyle \forall x \in A \cap B: \exists W \in \mathcal A \subseteq \mathcal B: x \in W \subseteq U \cap V$

2 implies 1
Let $U, V \in \mathcal B$.

Define the set:
 * $\displaystyle \mathcal A = \left\{{W \in \mathcal B: W \subseteq U \cap V}\right\} \subseteq \mathcal B$

By Union is Smallest Superset: General Result:
 * $\displaystyle \bigcup \mathcal A \subseteq U \cap V$

By hypothesis:
 * $\displaystyle \forall x \in U \cap V: \exists W \in \mathcal A: x \in W$

Thus $\displaystyle U \cap V \subseteq \bigcup \mathcal A$

By definition of set equality:
 * $\displaystyle U \cap V = \bigcup \mathcal A$