Particular Affirmative and Universal Negative are Contradictory

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

Then $\mathbf I$ and $\mathbf E$ are contradictory.

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

Proof
The argument reverses:

The result follows by definition of contradictory.