Proof of the Ultrametric Quality of $p$-adics

Question

We define the $p$-adic ordinal (or valuation) of $x \in \mathbb{Z}$ to be, $$ \operatorname{ord}_p (x) := \max \{ r: p^r \mid x \} $$ and the $p$-adic norm of $x$ to be, $$ |x|_p := \begin{cases} p^{-\operatorname{ord}_p (x)} & \text{if } x \ne 0 \\ p^{-\infty} = 0 & \text{if } x = 0 \end{cases} $$ I'm reading through a proof showing that this norm satisfies the ultra-metric inequality, $$|x+y|_p \le \max \{ |x|_p, |y|_p \}$$ (where we have an equality only when $|x|_p \ne |y|_p$), but I'm not understanding the final step. I do understand that $$ |x+y|_p = p^{-\operatorname{ord}_p (x+y)} \le p^{-\min \{ \operatorname{ord}_p (x), \: \operatorname{ord}_p (y) \}}, $$ but the author says that this implies that $|x+y|_p \le \max \{ |x|_p, |y|_p \}$. Unfortunately, I don't see why that is. I tried assuming (WLOG) that $\operatorname{ord}_p(x)$ was the minimum and simplifying the expression, but it didn't get anywhere. Could someone clarify?