Generating Finite Partition Preserves Order

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\BB, \CC \subseteq \Sigma$ be finite sub-$\sigma$-algebras.

Then:
 * $\BB\subseteq\CC\iff\map\xi\BB\le\map\xi\CC$

where:
 * $\map\xi\cdot$ denotes the generated finite partition
 * $\le$ denotes the order by refinement of partition