It is written in wikipedia: https://en.wikipedia.org/wiki/Dedekind-infinite_set
It is not provable (in ZF without the AC) that dual Dedekind-infinity implies that A is Dedekind-infinite. (For example, if B is an infinite but Dedekind-finite set, and A is the set of finite one-to-one sequences from B, then "drop the last element" is a surjective but not injective function from A to A, yet A is Dedekind finite.)
A is set of all finite subsets of B. Hence it has to be infinite and of same cardinality as of A. But 'drop the last elemenet' but which element.(For this we need choosing map from [0,n] to elements of A but then it requires AC but then these two definitions become equivalent.)
And why A is Dedekind finite.