Question on Proof of Set of Sets Proposition

Question

I am trying to prove the following statement regarding a set of sets $\mathscr{A}$:

$$x\notin \bigcap \mathscr{A} \; \text{iff there exists A} \in \mathscr{A} \; \text{such that} \; x \notin A$$

Here is what I have so far:

$$x\notin\bigcap \mathscr{A}$$ $$\text{if it is not the case that} \;x \in \bigcap \mathscr{A} $$ $$\text{if it is not the case that} \; x \in A \; \text{for all A} \in \mathscr{A}$$

I am a bit confused now though. Couldn't there be some $x \in A$ such that $A \in \mathscr{A}$, but that x does not belong to all sets in $\mathscr{A}$? In which case, wouldn't this then mean $x\notin\cap \mathscr{A}$?

Answer 1

You have managed to confuse yourself by putting the quantifier last. When you say "It is not the case that $x\in A$ for all $A$" it is unclear whether you mean

It is not the case that ($x\in A$ for all $A$).

or

(It is not the case that $x\in A$) for all $A$.

These are different meanings and you want only one of them. The confusion goes away if you follow the usual mathematical syntax (which is this way for a reason!) and put everything that modifies a single claim, including quantifiers and negations, before that claim. Then you get

It is not the case that for all $A$ we have $x\in A$.

which is unambiguous, and hopefully known to be equivalent to

There is an $A$ such that it is not the case that $x\in A$.