What is the correct quantifier order for 'almost everywhere' statements? For an expression like $P[X=Y]=1$ what is the correct way to unpack the expression into quantifier-like language?

Supposing X and Y are random variables with domain $\Omega$, is this correct?
For almost every $\omega\in\Omega$, $X(\omega)=Y(\omega)$.

Also, is it correct to think of expressions like 'almost every' as quantifiers?
We work on the space $(\Omega,\mathcal{F},P)$.

$P(X=Y)=1$ means $X$ and $Y$ are equal a.s. which can be written that way : $\exists A\in \mathcal{F},P(A)=0, \forall \omega \in A, X(\omega)\neq Y(\omega) $. (Here, just chose $A=(X\neq Y)$).
We work on the space $(\Omega,\mathcal{F},P)$.

$P(X=Y)=1$ means $X$ and $Y$ are equal a.s. which can be written that way : $\exists A\in \mathcal{F},P(A)=0, \forall \omega \in A, X(\omega)\neq Y(\omega) $. (Here, just chose $A=(X\neq Y)$).