Doubt about $p$-th root of unity

Question

Let $p\in\mathbb{Z}$ be an odd prime. Let $\zeta_p$ be a primitive $p$-th root of unity. Let $K=\mathbb{Q}(\zeta_p)$. Let $O_K$ be the ring of algebraic integrs. I am trying to prove that for any $2\leq i\leq p-1$ $$1+\zeta_p+\dots+\zeta_p^{i-1}\notin \langle1-\zeta_p\rangle$$ in $O_K$. How to prove this ?