Let $φ$ be purely inseparable isogeny of elliptic curves. Then, dual isogeny of $φ$ is always purely inseparable?
Background
Super singular elliptic curve over a field of characteristic $p$ is defined as dual of $p^r$-Frobenius is purely inseparable. But why this definition? Just like ''$E$ is super singular if it's $p^r$-frobenius is purely inseparable , is not equivalent to the definition of usual supers singular curve?
Certainly $\varphi$ being purely inseparable does not imply that its dual is as well.
Take for example an elliptic curve $E$ over $\overline{\mathbb{F}}_p$. Then we have the Frobenius $F:E\to E^{(p)}$ and its dual is called the Verschiebung $V:E^{(p)}\to E$. In fact if your elliptic curve is ordinary, the Verschiebung will be even étale (and not purely inseparable).
As the Frobenius will always be purely inseparable, your definition will not yield any condition. On the other hand requiring that the dual is purely inseparable is a proper condition and it will actually ensure that $$E[p](\overline{\mathbb{F}}_p)=0$$ as opposed to $$E[p](\overline{\mathbb{F}}_p)=\mathbb{Z}/p.$$