suppose that Y = Z - X is independent of Z and of X. Show that Y is constant.

my work so far:
want to prove V(Y) = 0
$\displaystyle V(Y) = V(X) + V(Z) -2Cov(Z,X)$
$\displaystyle = V(X) + V(Z) -2(V(Z+X)-V(X)-V(Z))$
$\displaystyle = 2V(X) + 2V(Z) - V(Z + X)$

and then I'm stuck. Any help appreciated.