Universal Affirmative and Particular Negative are Contradictory

Theorem
Consider the categorical statements:

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

Using the symbology of predicate logic:
 * $\neg \paren {\paren {\forall x: \map S x \implies \map P x} \iff \paren {\exists x: \map S x \land \neg \map P x} }$

Proof
The argument reverses:

The result follows by definition of contradictory.