What would be the proof that inner models are transitive?
Does it somehow use transitiveness of the model that they are compared to?
What would be the proof that inner models are transitive?
Does it somehow use transitiveness of the model that they are compared to?
It is part of the standard definition of an inner model. So it is quite trivial.
If it is not transitive not that we use $\in$ which is extensional and well-founded so it can be collapsed to a transitive class.
Generally speaking, inner models are substructure of the universe, as such we require them to have the same $\in$ relation as the "real" one. Then if they are not transitive they are well-founded and extensional (as those are properties of $\in$) and therefore isomorphic to a unique transitive class.