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What would be the proof that inner models are transitive?

Does it somehow use transitiveness of the model that they are compared to?

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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.

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