Basis Condition for Coarser Topology/Corollary 2

Theorem
Let $S$ be a set.

Let $\BB_1$ and $\BB_2$ be two bases on $S$.

Let $\tau_1$ and $\tau_2$ be the topologies generated by $\BB_1$ and $\BB_2$ respectively.

If $\BB_1 \subseteq \BB_2$ then $\tau_1$ is coarser than $\tau_2$.