Sentences of First Order Logic and its complement set.

Question

Let $\Delta$ be a set of FO sentences over signature $\Sigma$ and let $\Delta$ is unsatisfable. Does it mean that $\Delta'$ is satisfable? $\Delta' = \{x | \text{ x is a sentence FO }, x \not \in \Delta\}$ Why?

Answer 1

How about $\Delta = \{\forall x. x = x, \exists x . x \neq x\}$ which is clearly unsatisfiable, then we have $\forall x. \forall y. x = y \in \Delta'$ and $\exists x . \exists y. x \neq y \in \Delta'$ and so $\Delta'$ is also unsatisfiable.