Prove or disprove $span({v_0, v_1,...,v_n}) = span({v_0, v_1 - v_0, ..., v_n-v_0})$

Question

As the title says. I am under the impression that the right hand side refers to an equivalence class (since we can show that the set ${v_0, v_1-v_0, ..., v_n - v_0}$ is a subspace and therefore the differences are contained in that subspace), and that if we take $v_0$ to be the zero vector, which we know is an element of all vector spaces, then we show equality. However, I'm not sure if this is the right approach.