Alternative way for expressing "No student in this class has sent e-mail to exactly two other students in this class."

Question

In my HW, I can see that the following statement

"No student in this class has sent e-mail to exactly two other students in this class."

can be expressed as follows

$$¬∃x∃y∃z(y \neq z ∧x \neq y ∧x \neq z ∧ ∀w(w \neq x ⇒ (E(x, w) ⇔ (w = y ∨ w = z))))$$

But my initial thought was this:

$$\neg \exists x \exists!y\exists!z((x \neq z) \land (x \neq y)\land(y \neq z)\land Email(x,y) \land Email(x,z))$$

Is my initial thought incorrect, and if so what is wrong with it?

Answer 1

It's not correct. The first thing that strikes me is that if $y$ is unique, then how can you say $z$ is unique when it has the same property as $y$? Clearly they're not unique. The pair is what you want unique.