No Valid Categorical Syllogism with Particular Premise has Universal Conclusion

Theorem
Let $Q$ be a valid categorical syllogism.

Let one of the premises of $Q$ be particular.

Then the conclusion of $Q$ is also particular.

Proof
From No Valid Categorical Syllogism contains two Particular Premises, at least one premises of $Q$ has to be universal.

Let the other premise of $Q$ be particular.

Suppose the conclusion of $Q$ be $\mathbf{A} \left({S, P}\right)$.

Then by definition the secondary term $S$ of $Q$ is distributed.

By Distributed Term of Conclusion of Valid Categorical Syllogism is Distributed in Premise it follows that $S$ is distributed in the minor premise $\text{Min}$ of $Q$.

By Middle Term of Valid Categorical Syllogism is Distributed at least Once, the middle term $M$ of $Q$ needs to be distributed in either the major premise $\text{Maj}$ or minor premise $\text{Min}$ of $Q$.

Suppose $M$ were distributed in $\text{Min}$.

Then $\text{Min}$ would have both its terms distributed.

So by definition $\text{Min}$ would be the universal negative $\text{E}$.

But from Conclusion of Valid Categorical Syllogism is Negative iff one Premise is Negative this is not the case.

Suppose $M$ were distributed in $\text{Maj}$.