By # 4232, we have $C_p^k\equiv 0\pmod{p}$ for all the positive integer $k$ less than $p$. Therefore, we have $$\begin{array}{ll} (x+y)^p &= x^p + C_p^1x^{p-1}y + \cdots + C_p^{p-1}xy^{p-1}+y^p \\ &\equiv x^p + 0 +\cdots + 0 + y^p \\ &\equiv x^p + y^p \pmod{p}\end{array}$$