A property of the space $W_{0}^{1,p}$ concerning $u\le v \in W^{1,p}(\Omega)$ on $\partial \Omega$ In the weak maximum principle for weak solution.

In Gilbarg-Trudinger Section 8.1. The Weak Maximum Principle, they define $u\le v \in W^{1,2}(\Omega)$ on $\partial \Omega$ by $(u - v)^+\in W_0^{1,2}(\Omega)$.

My question is: does the following statement hold? Given $0\leqslant f \in W_0^{1,2}$ and $g \in W^{1,2}\text{ s.t. }0\leqslant g\leqslant f$, we have $g\in W_0^{1,2}$.

My interest lies in proving some analogous common properties of $\le$ on $\partial\Omega$. For example, if the above statement is true, then $u\le v$ and $v\le w$ on $\partial \Omega$ would imply $u\le w$ on $\partial \Omega$, by noticing $a^++ b^+ \ge (a+b)^+$

PS. I'm now aware that there's an answer to my question here. But there, $W^{1,p}$ seems to be defined by approximation of $C^\infty$ functions under $W^{1,p}$ norm. What if $W^{k,p}$ is understood by having $L^p$ weak derivatives up to $k$th order.