Let $\omega \in \Omega$ be an outcome such that $X(\omega) \in E$. If $X(\omega) = Y(\omega)$, then also $Y(\omega) \in E$ and else $X(\omega) \neq Y(\omega)$. Since $\omega \in \Omega$ was arbitrary, we have the inclusion
\begin{equation}
\{ X \in E \}
\subseteq \{ X \neq Y \} \cup \{ Y \in E \}
\end{equation}
We can now apply results
to conclude
\begin{equation}
\begin{split}
\mathbb{P}(X \in E)
& \leq \mathbb{P}(X \neq Y \text{ or } Y \in E)
& \leq \mathbb{P}(X \neq Y) + \mathbb{P}(Y \in E)
\end{split}
\end{equation}
$\square$
