There does not exist rational numbers $x$ and $y$ such that $x^y$ is a positive integer and $y^x$ is a negative integer

Question

I want to prove or disprove:

There does not exist rational numbers $x$ and $y$ such that $x^y$ is a positive integer and $y^x$ is a negative integer.

For the integers $-3$ and $4$, $(-3)^4 = 81$ and $4^{-3} = 1/64$.

There seems to be no rational numbers satisfying the condition.

Can you comment on this?

Answer 1

If $y^x$ is a negative integer, then $y$ must be a negative integer (or else $y^x > 0$) and $x$ must be a positive integer (or else $0 < |y^x| < 1$).

But then $0 < x^y < 1$, so $x^y$ is not a positive integer.