Union of Connected Sets with Non-Empty Intersections is Connected/Corollary/Proof 1

Proof
Let $C \in \mathcal A$.

From Space with Connected Intersection has Connected Union applied to $B$ and $C$, the union $B \cup C$ is connected.

Thus the set $\tilde{\mathcal A} = \left\{{B \cup C: C \in \mathcal A}\right\}$ satisfies the conditions of the theorem.

Hence the result.