Must the degenerate simplices of a nondegenerate simplex be different?

Question

Let $\sigma$ be a nondegenerate $n$-simplex in a simplicial set. Does it follow that the degenerate simplices $s_0(\sigma)$ and $s_1(\sigma)$ are different?

For instance, consider $\Delta^2$, the triangle. Indeed, the nondegenerate 1-simplex $(0,1)$ is mapped by $s_0$ and $s_1$ to $(0,0,1)$ and $(0, 1, 1)$, respectively. I wonder whether this generalizes to all simplicial sets.

Answer 1

Suppose we have an $n$-simplex $\sigma$ such that $s_i (\sigma) = s_{i+1} (\sigma)$. Then, $$\sigma = d_i (s_i (\sigma)) = d_i (s_{i+1} (\sigma)) = s_i (d_i (\sigma))$$ so $\sigma$ is degenerate.