Strictly dominant strategy equilibrium uniqueness proof

My attempt is:

Suppose $$s^D$$ is a strictly dominant strategy equilibrium. Then by definition $$s_i^D$$ $$\in$$ $$S_i$$ is a strict dominant strategy for all $$i \in N$$. This means that all $$s_{-i}$$ are strictly dominated by $$s_i^D$$, and hence there cannot exist any other strictly dominant strategy equilibrium, because that would mean that $$s^D$$ is strictly dominated by some $$s_{-i} \in S_{-i}$$, which is a contradiction.

Could someone tell me whether this is correct, and if not then hint me?