equivalence between a set and the subset of its subset

batballbat

Is it true that: If A is not equivalent to its subset A1. Then A is not equivalent to any subset of A1?

Norwegian

You should consider also informing us about the relevant equivalence relation, as the truth or falsity of your statement heavily depends on that information.

For example, if you for your eq.rel. use existence of a bijection, the statement is true, but other relations, like having same parity, will render your statement false.

batballbat

actually this is trivial. I just learnt all this the complicated way, .i.e. proving cantor bernstein without the well ordering theorem.