Let $\Omega\subset\mathbb{R}^n$ a open bounded set. The Dirichlet laplacian can be defined via it's closed semi-bounded form on $H^1_0(\Omega)$. The fact that it's spectrum is discrete is as far as I can tell proven by that fact that the embedding $H_0^1(\Omega)\rightarrow L^2(\Omega)$ is compact and that the spectrum is discrete if and only if the embedding $H_0^1(\Omega)=(D(q),\lVert\cdot\lVert_q)\rightarrow L^2(\Omega)$ is compact. Where $q$ is the associated form and $D(q)$ is the form domain.
I search for quite some time but didn't find a proof for the second claim. I'd appreciate hints on the proof itself and references very much!