Proof of "Entropy of preparation" in Von Neumann entropy

How should I prove this?

From John Preskill's quantum computation & quantum information lecture notes(chapter 5)

If a pure state is drawn randomly from the ensemble{|φx〉,px}, so that the density matrix is ρ = ∑pxx〉<φx|
Then, H(X)≥S(ρ)
where H stands for Shannon entropy of probability {px} and S stands for Von Neumann entropy.

In short, you express $|\phi_{x}\rangle$ in terms of the eigenstates of the density matrix $\rho$. Since the entropy of a mixture of states is larger than the corresponding mixture of entropies, you can eventually find:
$S(\rho)\leq H(X)+\sum_{x}p_{x}S(\rho_{x})$