We have \begin{equation} \begin{split} \mathbb{P}(\{ X \in E \} \cap \{ X = Y \}) & = \mathbb{P} \left( \{ \omega \in \Omega : X(\omega) \in E \} \cap \{ \omega \in \Omega : X(\omega) = Y(\omega) \} \right) \\ & = \mathbb{P} \left( \{ \omega \in \Omega : X(\omega) \in E \text{ and } X(\omega) = Y(\omega) \} \right) \\ & = \mathbb{P} \left( \{ \omega \in \Omega : X(\omega) = Y(\omega) \in E \text{ and } X(\omega) = Y(\omega) \} \right) \\ & = \mathbb{P} \left( \{ \omega \in \Omega : Y(\omega) \in E \text{ and } X(\omega) = Y(\omega) \} \right) \\ & = \mathbb{P} \left( \{ \omega \in \Omega : Y(\omega) \in E \} \cap \{ \omega \in \Omega : X(\omega) = Y(\omega) \} \right) \\ & = \mathbb{P}(\{ Y \in E \} \cap \{ X = Y \}) \end{split} \end{equation} $\square$