Maybe I'm just need to buff up on my logic notation, but I don't fully understand the following:
$$\exists y\forall z \left(\exists w(z\in w\wedge w\in x)\implies z\in y\right)$$
How should I unravel these statements generally? Starting with the innermost parens? As best I can tell the part starting with $\exists w$ means that there exists some $w$ such that $z$ is an element of $w$ and $w$ is an element $x$ which implies that $z$ is an element of $y$. But I dont understand how to parse $\exists y \forall z$ type statements (i.e. when they're up against each other like that). How do I even read that? "There's some element $y$ for all $z$'s"?
As you can tell, I'm generally confused. Can someone provide some guidance?