dimbulb, the proof is tricky, but there are good expositions. see e.g.
Yiannis Moschovakis
Notes on set theory
Springer, 1994.
The trick is to keep track of the path of an element when the two injections are followed. If you want to find a proof by yourself, try distinguishing those points on S that have infinite, and those that have finite such trajectories.
BTW, I don't see where your reformulation helps?
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