I need help understanding a certain part of the following proof.

Theorem: (Monotone Class Theorem)

Let $\displaystyle \mathcal{C} $ be a class of subsets of $\displaystyle \Omega$, closed under finite intersections and containing $\displaystyle \Omega$. Let $\displaystyle \mathcal{B}$ be the smallest class containing $\displaystyle \mathcal{C}$ which is closed under increasing limits and by difference. Then $\displaystyle \mathcal{B} = \sigma(\mathcal{C})$.

Proof:

The intersection of classes of sets closed under increasing limits and differences is a class of that type. So by taking the intersection of all such classes, there always exists a smallest class containing $\displaystyle \mathcal{C} $ which is closed under increasing limits and by differences. For each set $\displaystyle B $, denote

$\displaystyle \mathcal{B}_{B}$ to be the collection of sets $\displaystyle A$ such that $\displaystyle A \in \mathcal{B}$ and $\displaystyle A \cap B \in \mathcal{B}$.

Clearly $\displaystyle \mathcal{B}_{B}$ is closed under increasing limits and by difference.

Let $\displaystyle B \in \mathcal{C}$; for each $\displaystyle C \in \mathcal{C}$ one has

$\displaystyle B \cap C \in \mathcal{C} \subset \mathcal{B}$ and $\displaystyle C \in \mathcal{B}$, so $\displaystyle C \in \mathcal{B}_{B}$.

HERE THE PROBLEM STARTS

Now let $\displaystyle B \in \mathcal{B}$. For each $\displaystyle C \in \mathcal{C}$ we have $\displaystyle B \in \mathcal{B}_{C}$, and because of the preceeding

$\displaystyle B \cap C \in \mathcal{B}$, hence $\displaystyle C \in \mathcal{B}_{B}$, whence $\displaystyle \mathcal{C} \subset \mathcal{B}_{B} \subset \mathcal{B}$, hence $\displaystyle \mathcal{B} = \mathcal{B}_{B}$.

The rest of the proof is just concluding that $\displaystyle \mathcal{B}$ is a $\displaystyle \sigma$-algebra.

Back to the problem part. Why must $\displaystyle B \in \mathcal{B}_{C}$?

It's easy to see when $\displaystyle B \in \mathcal{C}$, because $\displaystyle \mathcal{C}$ is closed under finite intersections.