De Morgan's Laws (Predicate Logic)/Assertion of Universality

Theorem
Let $\forall$ and $\exists$ denote the universal quantifier and existential quantifier respectively.

Then:
 * $\forall x: \map P x \dashv \vdash \neg \paren {\exists x: \neg \map P x}$
 * If everything is, there exists nothing that is not.