If $I$, $J$ are ideals of commutative Noetherian ring with unity, and $I\subsetneq J$, then is $\operatorname{ht} I < \operatorname{ht} J$?

Question

Let $I$, $J$ be ideals in the commutative Noetherian ring $R$ (with unity) such that $I\subset J$ (strict subset). Must it always be the case that $\operatorname{ht} I < \operatorname{ht} J$ ?

If $I$ and $J$ both are prime ideals, then this is true. But here given $I$ and $J$ any arbitrary ideal of $R$.

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Edit: I think I got the answer. Answer will be NO.

Take $R=K[x, y]$ where $K$ is a field. Take Ideal $I= <x^2, xy>$ and $J= \sqrt{I} = <x, y>$, then $\operatorname{ht} I = \operatorname{ht} J$ but $I \subset J$.

Am I correct?