A set which is not isomorphic to it's proper subset is finite

set-theory solution-verification elementary-set-theory

To be clear, you should say that you are proving the statement by induction on the cardinality of $A$. Your argument then works, assuming that you have already proved that cardinalities are well-ordered, so you can do induction on them. Given the nature of the question though it seems unlikely that you have that result available to you, since if you already had the full theory of ordinals and cardinals developed the result you are talking about would be rather trivial.

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