If $X$ elliptic surface, $C\subset X$ is a curve with $C\cdot (-K_X)=0$, then $C=-K_X$

Question

Let $X$ be a rational, relatively minimal elliptic surface over $\Bbb{C}$.

We know that $-K_X$ is linearly equivalent to any fiber of the elliptic fibration $\pi:X\to\Bbb{P}^1$.

Now let $C\subset X$ be an irreducible curve such that $C\cdot (-K_X)=0$. Is it true that $C$ must be a fiber?

Here's my attempt: if $P\in C$ is any point, then the fiber $F:=\pi^{-1}(\pi(P))$ intersects $C$ in at least one point, namely $P$. So if $C\neq F$, then $C\cdot F>0$, contradicting $C\cdot F=C\cdot (-K_X)=0$.

It seems so elementary I thought I'd better ask. Thank you!