I am having a very difficult time understanding this formula in set-builder notation.
$\forall x \forall y \exists z \forall u \left ( u \in z \Leftrightarrow \left (z=x \lor u=y \right) \right)$
For two sets $x,y$ there exists a set $z$ such that for all sets $u$, $u \in z$ iff $u \in x$ or $y \in z$.
Every element of $z$ must be an element of $x$ or $z$ is just the set $y$.
This is the set that contains elements of $x$ and the set $y$. I am not so sure what to make of this.