Universal Affirmative and Particular Negative are Contradictory

Theorem
Consider the categorical statements:
 * $\mathbf A: \quad$ The universal affirmative: $\forall x: S \left({x}\right) \implies P \left({x}\right)$
 * $\mathbf O: \quad$ The particular negative: $\exists x: S \left({x}\right) \land \neg P \left({x}\right)$

Then $\mathbf A$ and $\mathbf O$ are contradictory.

Using the symbology of predicate logic:
 * $\neg \left({\left({\forall x: S \left({x}\right) \implies P \left({x}\right)}\right) \iff \left({\exists x: S \left({x}\right) \land \neg P \left({x}\right)}\right)}\right)$

Proof
The argument reverses:

The result follows by definition of contradictory.