Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X, Y : \Omega \to \Xi$ are each a D202: Random variable on $P$
(ii)	$\{ X \in E \}, \{ Y \in E \} \in \mathcal{F}$ are each an D1716: Event in $P$

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

(i)	R2090: Isotonicity of probability measure
(ii)	R4739: Binary subadditivity of probability measure

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} $\square$