probability - Clarification of "$in$" in ${ omegain Omega: X(omega)leq x } in mathcal{F}$ for a random variable

I have a basic question regarding the definition of a random variable. Probability and Random Processes (Grimmett and Stirzaker) have the following:

A random variable is a function $X:\Omega\rightarrow \mathbb{R}$ with the property that
$\{ \omega\in \Omega: X(\omega)\leq x \} \in \mathcal{F}$
for each $x\in \mathbb{R}$. Such a function is said to be $\mathcal{F}$-measurable.

Q1: Because of the curly brackets I guess $\{\omega\in \Omega: X(\omega)\leq x\}$ is a set, right?

Q2: I know if $a$ is an element of the set $A$ we write $a\in A$. But if a set $B$ is a subset of a set $C$, we write $B\subset C$ and not $B\in C$.

So if $\{\omega\in \Omega: X(\omega)\leq x\}$ is a set shouldn't we use "$\subset$" instead of "$\in$", i.e.
$$\{ \omega\in \Omega: X(\omega)\leq x \} \subset \mathcal{F} \qquad ? \tag 1$$