Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $\{E_i\}_{i = 1}^N,$ with $E_i \in\mathcal{F}$ be a set of events and let $i(X)$ be a R.V. assuming values in $\{1,...,N\}$

Is there a way to bound the following quantity?

$$\mathbb{P}\left[\bigcup_{i\in[N]: i \neq i(X)} E_i\right].$$

I am looking in an upper-bound that resemble the union bound: in fact, if the union would not depend on the R.V. $X$ we may use the union bound in the following way

$$\mathbb{P}\left[\bigcup_{i\in[N]} E_i\right] \leq \sum_{i\in[N]}\mathbb{P}\left[ E_i\right].$$