Connectedness of complement of intersection of two balls

Let $B_r(x_0)\subset \mathcal{R}^n$ be an open ball with radious $r>0$ and centered at $x_0$. Let $y_0\in \partial \overline{B}_r(x_0)$. I want to argue: there exists some $\delta>0$ such that both $B_\delta(y_0)\cap B_r(x_0)$ and $B_\delta(y_0)\setminus B_r(x_0)$ are connected. Is the statement true? If so, how do I make the argument rigorous? If not, could you please provide a counter example?

It is clear that $B_\delta(y_0)\cap B_r(x_0)$ is connected because intersection of two convex sets is convex and hence connected. I am having a hard time proving $B_\delta(y_0)\setminus B_r(x_0)$ is connected. From drawing little diagrams in $\mathcal{R}^2$, my intuition tells me it should be. Any help would be greatly appreciated.