Is nonnegative $C_c^\infty$ dense in nonnegative $H_0^1$?

I have encountered the same problem as Replacing $C_c^\infty$ by $H_0^1$ in the definition of weak subsolution.

I wonder how to prove this statement there.

If $\phi\in H_0^1(\Omega)$ and $\phi \geq 0$, then there is a sequence $(\phi^n)$ in $C_0^\infty$ such that $\phi^n\geq 0$ and $\phi^n\to \phi$ in $H^1(\Omega)$.

Since from the definition of $H_0^1(\Omega)$, we only know that there exists $(\phi^n)$ in $C_0^\infty$ and $\phi^n\to \phi$ in $H^1(\Omega)$. What about the nonnegativity of $\phi^n$?