By definition, $I$ is a Cartesian product of some basic real intervals $I_1, \dots, I_N$. Let $a_1, b_1, \dots, a_N, b_N$ be the endpoints of $I_1, \dots, I_N$, respectively. Result
R2973: The four classes of real intervals under reflection states that now $-b_1, - a_1, \dots, -b_N, -a_N$ are the endpoints of the reflected intervals $- I_1, \dots, - I_N$, respectively. Applying result
R2969: Euclidean real Cartesian product of scaled sets now yields
\begin{equation}
\begin{split}
\mathsf{Vol}(- I) & = \mathsf{Vol} \Big( - \prod_{n = 1}^N I_n \Big) \\
& = \mathsf{Vol} \Big( \prod_{n = 1}^N -I_n \Big) \\
& = \prod_{n = 1}^N |(-b_n) - (-a_n)| \\
& = \prod_{n = 1}^N |b_n - a_n| \\
& = \mathsf{Vol} \Big( \prod_{n = 1}^N I_n \Big) \\
& = \mathsf{Vol}(I)
\end{split}
\end{equation}
$\square$