How prove $E|X|^p<\infty$ and $E|Y|^p<\infty$, if $E|X+Y|^p<\infty$

Let two random variables $X$ and $Y$ be independent of each other, for some $p>0$, we have $$E|X+Y|^p<\infty$$ Show that $$E|X|^p<\infty,\quad E|Y|^p<\infty$$

My try: I know Minkowski inequality

$$E|X+Y|^p \leq E|X|^p + E|Y|^p, \tag{$0<p<1$}$$

But this inequality here is not useful, so how prove it? Thank you very much!

We have $$\iint_{\mathbb R^2}|x+y|^p\mathrm dP_X(x)\mathrm dP_Y(y)=\mathbb E|X+Y|^p\lt \infty,$$ hence for $P_Y$-almost every $y$, $\int_\mathbb R|x+y|^p\mathrm dP_X(x)<\infty$. We deduce that $\mathbb E|X|^p<\infty$ by triangular inequality. Indeed, there is a constant $C_p$ such that $|a+b|^p\leqslant C_p(|a|^p+|b|^p)$, hence $$\int_\mathbb R|x|^p\mathrm dP_X(x)\leqslant C_p\int_\mathbb R|x+y|^p\mathrm dP_X(x)+C_p|y|^p\lt\infty$$ for a $y$ such that $\int_\mathbb R|x+y|^p\mathrm dP_X(x)\lt\infty$.