Actually, the usual proof of the Cantor-Schröder-Bernstein Theorem does not use the Axiom of Infinity (nor the Axiom of Powersets).
By the usual proof, I mean the one found on Wikipedia, for example. Using the notation from that proof, the main point of contention is whether we can form the sets Cn and C=bigcupin=0nftyCn. These sets exist by comprehension:
C0=xinA:forallyinB,(xneqg(y))
Cn=xinA:existss,(s:0,dots,ntoAlands(0)inC0land(foralli<n)(s(i+1)=g(f(s(i))))lands(n)=x,
where abbreviations such as s:0,dots,ntoA should be replaced by the equivalent (bounded) formulas in the language of set theory. The definition of Cn is uniform in n, and so
C=xinA:existsn,(mathrmFinOrd(n)landxinCn),
where xinCn should be replaced by the above definition and mathrmFinOrd(n) is an abbreviation for "n is zero or a successor ordinal and every element of n is zero or a successor ordinal." An alternate definition of C is
C=xinA:forallD,(C0subseteqDsubseteqAlandg[f[D]]subseteqDtoxinD),
which shows that C is Delta1-definable. The rest of the proof uses induction on finite ordinals, but since the definition of the sets Cn is uniform these are special cases of transfinite induction.
In conclusion, it looks like the proof could work in Kripke-Platek Set Theory — which has neither the Axiom of Infinity nor the Axiom of Powersets — provided that the two definitions of C given above are provably equivalent. I haven't tried to check whether the two definitions are provably equivalent but, in any case, the proof can be carried out in Kripke-Platek Set Theory with Sigma1-Comprehension.
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