Since w ∈ S(u), for every vertex x ∈ S(u), there is a directed path from w to x and from x to w. Similarly, since w ∈ S(v), for every vertex y ∈ S(v), there is a directed path from w to y and from y to w.

Putting those paths together, there is a directed path from x to y and from y to x (going through w) for all x ∈ S(u) and y ∈ S(v). That means x and y are in the same strongly connected component.

Since x is an arbitrary member of S(u) and y is an arbitrary member of S(v), S(u) must be the same as S(v).