Equivalence between a set and the subset of its subset

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Is it true that: If A is not equivalent to its subset A1. Then A is not equivalent to any subset of A1?
 
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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.
 
actually this is trivial. I just learned all this the complicated way, .i.e. proving cantor bernstein without the well ordering theorem.
 

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