In my effort to express "everything is different from everything else", my first approach was to try "There are exactly n distinct things". The hope was that I could generalise the statement by adding a universal. For example:
$\forall x (x=x)$ identity
$(\exists x \exists y \exists z \neg(x=y) \land \neg(x=y) \land \neg(x=z) \land \neg(y=z))$ for ($n=3$).
Which seems to express difference using negated identity.
But, I could get no further.
What does "Everything is different from everything else'' mean. It could be saying: take anything thing and any other thing: then they are different things. Which is trivially true:
$$\forall x\forall y(x \neq y \to x \neq y)$$
Or something more substantial might be meant: given two distinct things there is some intrinsic property which distinguishes them (the contrapositive of a version of Leibniz's disputable principle of the identity of indiscernibles). But we'll need more than pure FOL to express this.