Why is the intersection of the empty set the universe? I haven't found an answer yet that I can make complete sense of, so I'm asking again to try and clear it up if I have any misconceptions. 
If one defines $\cap_{i\in I}\alpha_i = \{x:\forall i\in I, x\in\alpha_i\}$ then apparently if $I = \emptyset$ this definition yields the absolute universe. This is just stated as if it is clear why, though I cannot see why. If $i\notin I \forall i$ then there is no set $\alpha_i$ for any $x$ to be a member of...?
I know I must be misreading this, but I can't see how by so much...
Edit: Let this intersection be $Z$, for convenience.

Any statement of the form $\forall i\in I, ...$ is true if $I$ is empty, so in particular the statement $\forall i\in I, x\in \alpha_i$ is true for all $x$. Perhaps the easiest way to see this, is that the negation of this statement is: $\exists i\in I$ such that $x\notin \alpha_i$, which is clearly false if are no $i$ to begin with.
