Since $E_0, E_1, E_2, \ldots$ are all stationary in $M$, applying
R4470: Inverse image of countable union is union of inverse images, we have
\begin{equation}
T^{-1} \left( \bigcup_{n \in \mathbb{N}} E_n \right)
= \bigcup_{n \in \mathbb{N}} T^{-1} E_n
= \bigcup_{n \in \mathbb{N}} E_n
\end{equation}
$\square$
