Does a model that satisfies both of these formulas really exists?

Question

Find a model that satisfies

$\forall x \neg{ R(x,x) }$

and

$\forall x \forall y R(x,y)$

Where $R$ is a binary function.

It seems to me that $\forall x \neg{ R(x,x) }$ is part of the set that is defined by $\forall x \forall y R(x,y)$

Answer 1

Clearly not (if we restrict ourselves to interpretations with non-empty domains).

From 1st premise we get $\lnot R(a,a)$ by UI, and by the same rule we get $R(a,a)$ from the second premise.