The Door plucks us from different realities, it's not a conspiracy to imagine it would bring us from different timelines as well - that's simple logic.

[There are any number of ways he could show his work, but the one that is most effective is also likely the one that is the most cutting for a Wammy's child - revealing secrets.]

My proof? [He smiles sharply, mouth closed.] I know your name. And I know that you know mine, because you can see it. And I know why.