Help with semi-formal logic

Question

How do I write semi-formally 'there are only 2 objects in the universe'? My hypothesis is: ∃x∃y(x≠y) Any ideas?

Answer 1

$\exists x \exists y \forall z ((x\neq y) \land ( (z=x) \lor (z=y))$

Answer 2

Define $a, b$ and $U$ such that $$a\ne b \land \forall x\in U:[x=a\lor x=b]$$

where $U$ is your universe.

Then recast your hypothesis as: $$\exists x\in U: \exists y\in U: x\ne y$$