I'd like to prove the following proposition.
"If Σ is an orientable surface and $y$ is a closed curve on Σ such that $i(x, y) = 0$ for every simple closed curve $x$. Then $y$ is either homotopically trivial or homotopic to a boundary curve or homotopic to a puncture."
where $i(x, y)$ is geometric intersection number of $x$ and $y$
$i.e.$ $i(x, y)$ is min{#$x$ $\cap$ $y$|$x\in[x]$,$y\in[y]$} ($[x]$ is free homotopy class of $x$ )
I have an idea which is "If $y$ is essential closed curve, then there is some loop $y'$ such that $i(y, y') >0$"
But I still can't prove it. Please give me some advice.