Let $A=\mathbb{C}[x_0,x_1,\dots,x_n]$ and $X=\operatorname{Proj}A$.
For any homogeneous ideal $I\subset A$, define the saturation $I^{\rm sat}:=\{f\in A\mid (x_0,\dots,x_n)^mf\subset I$ for some $m\ge0\}$.
$I$ is called a saturated ideal if $I=I^{\rm sat}$.
I know that there is a 1-1 correspondence between the saturated ideals of $A$ and closed subschemes of $X$. In particular, $\operatorname{Proj}(A/I^{\rm sat})=\operatorname{Proj}(A/J^{\rm sat})$ if and only if $I^{\rm sat}=J^{\rm sat}$.
My question is that
If $\operatorname{Proj}(A/I^{\rm sat})\subseteq \operatorname{Proj}(A/J^{\rm sat})$ (as closed subschemes) then $J^{\rm sat}\subseteq I^{\rm sat}$? Is $J^{\rm sat}\nsubseteq I^{\rm sat}$ possible?