Let $E/K$ be an elliptic curve over a number field, and $\mathfrak{p}$ a prime of good supersingular reduction. Let $p$ be the prime below $\mathfrak{p}$. I believe that the following is true, but I can't prove it, hence my asking here:

$E$ does not possess a $K$-rational $p$-isogeny.

I think this is true because, roughly, supersingular primes are ``rare'', and so too are rational isogenies, so asking for both at once is probably asking for too much. Googling has not helped me.