Let $M$ be a fine structural mouse and let $F\in M$ be such that $M\models$``$F$ is a total extender'' and $(M||\mathrm{ lh}(F),F)$ is a premouse. We show that it follows that $F\in\mathbb{ E}^M$, where $\mathbb{ E}^M$ is the extender sequence of $M$. We also prove generalizations of this fact. Let $M$ be a premouse with no largest cardinal and let $\Sigma$ be a sufficient iteration strategy for $M$. We prove that if $M$ knows enough of $\Sigma\upharpoonright M$ then $\mathbb{ E}^M$ is definable over the universe $\lfloor M\rfloor$ of $M$, so if also $\lfloor M\rfloor\models\mathrm{ ZFC}$ then $\lfloor M\rfloor\models$``$V=\mathrm{ HOD}$''. We show that this result applies in particular to $M=M_{ \mathrm{ nt}}|\lambda$, where $M_{ \mathrm{ nt}}$ is the least non-tame mouse and $\lambda$ is any limit cardinal of $M_{ \mathrm{ nt}}$. We also show that there is no iterable bicephalus $(N,E,F)$ for which $E$ is type $2$ and $F$ is type $1$ or $3$. As a corollary, we deduce a uniqueness property for maximal $L[\mathbb{ E}]$ constructions computed in iterable background universes.
Categories: math.LO