{$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$

Question

I have a question that it maybe simple but I can not understand why we have :

{$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$

Should I use some inequality? Or is it obvious from definition?

Thanks