Question about this proof of the Singleton Bound?

Question

I have the following proof of the Singleton Bound (screenshot taken directly from a lecture slide):

This is fairly straightforward to the point where it claims that $d(x,y) \leq d-1$.

However, the proof then goes on to say that since $d(x,y) \leq d-1$, the mapping $\pi : C \mapsto S^{n - (d-1)}$ is one-to-one (or injective). Why is this?

Answer 1

$x\neq y\in C$ implies $d(x,y)\geq d$ by the definition of an $(n,M,d)$ code. So $d(x,y)\leq d-1$ is a contradiction.