Particular Affirmative and Universal Negative are Contradictory

Theorem
Consider the categorical statements:
 * $\mathbf I: \quad$ The particular affirmative: $\exists x: \map S x \land \map P x$
 * $\mathbf E: \quad$ The universal negative: $\forall x: \map S x \implies \neg \map P x$

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

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

Proof
The argument reverses:

The result follows by definition of contradictory.