Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 2

Theorem
Let $S$ be a set.

Let $\BB$ be a synthetic basis on $S$.

Let $\tau$ be the topology on $S$ generated by the synthetic basis $\mathcal B$:
 * $\tau = \set{\bigcup \AA: \AA \subseteq \BB}$

Then:
 * $\forall U \subseteq S: U \in \tau \iff U = \bigcup \set {B \in \BB: B \subseteq U}$

Proof
Trivially, the reverse implication holds, as $\set{B \in \BB: B \subseteq U} \subseteq \BB$.

We now show that the forward implication holds.

Suppose that $U \in \tau$. Then, by definition:
 * $\exists \AA \subseteq \BB: U = \bigcup \AA$

By Union is Smallest Superset: General Result, we have:
 * $\forall B \in \AA: B \subseteq U$

By the definition of a subset, it follows that:
 * $\AA \subseteq \set{B \in \BB: B \subseteq U}$

From Union of Subset of Family is Subset of Union of Family:
 * $U = \bigcup \AA \subseteq \bigcup \set{B \in \BB: B \subseteq U}$

By Union is Smallest Superset: General Result:
 * $\bigcup \set{B \in \BB: B \subseteq U} \subseteq U$

By definition of set equality:
 * $U = \bigcup \set{B \in \BB: B \subseteq U}$