Open Balls whose Distance between Centers is Twice Radius are Disjoint

Theorem
Let $M = \struct {A, d}$ be a metric space.

Let $x, y \in A$ such that $\map d {x, y} = 2 r > 0$.

Let $\map {B_r} x$ denote the open $r$-ball of $x$ in $M$.

Then $\map {B_r} x$ and $\map {B_r} y$ are disjoint.

Proof
$\map {B_r} x \cap \map {B_r} y \ne \O$.

Then:

But this contradicts our initial assertion that $\map d {x, y} > 2 r$.

The result follows by Proof by Contradiction.