Set Theory and Beyond

I've already told you what the union of two sets is. To remind you, if $x$ and $y$ are sets, then $x\cup y$ is the set whose elements are exactly all the elements of $x$ together with all the elements of $y$. We established existence of this union set by appealing to the Axiom of Union. Two other basic set operations that you've probably heard about are intersection $\cap$ and set difference/minus $-$. However, these two are a bit different from the union and their existence follows from the Axiom of Comprehension. Let $x$ and $y$ be two sets. Their intersection should consist exactly of those sets which belong to both $x$ and $y$. We can say this also in another way: the elements of the intersection of $x$ and $y$ are exactly those elements of $x$ which are also elements of $y$. Maybe you've caught onto what I'm trying to suggest: the intersection of $x$ and $y$ is the set $$x\cap y:=\{z\in x:P_y(z)\}$$ where $P_y(z)$ is the proper...