Let us call a map $f: X \to Y$ between non-empty sets a "weak injection" if $f^{-1}(\{y\})\subseteq X$ is finite for every $y \in Y$.
Recall that the (Schroeder-)Cantor-Bernstein-Theorem (sometimes abbreviated by (CB)) states that if there are injections between two sets $X, Y$, then there is also a bijection between $X$ and $Y$. (CB) is a theorem of ${\sf (ZF)}$.
Consider the following statement:
(wiCB) If $X, Y$ are infinite sets such that there are weak injections between $X$ and $Y$, then there is a bijection between $X$ and $Y$.
Obviously, (wiCB) implies (CB). Can (wiCB) also be proved in ${\sf (ZF)}$?
Best Answer
No, it is not provable in $\mathsf{ZF}$.
It is consistent with $\mathsf{ZF}$ that there is a sequence of disjoint two-element sets whose union is not countable, i.e. $\vert A_i\vert=2$ but there is no bijection between $A:=\bigcup_{i\in\mathbb{N}}A_i$ and $\mathbb{N}$.
WLOG, $A\cap\mathbb{N}=\emptyset$. Now let $B_i=A_i\sqcup\{i\}$ and let $B=\bigcup_{i\in\mathbb{N}}B_i$. We have obvious weak injections $B\rightarrow\mathbb{N}$ and $\mathbb{N}\rightarrow B$, the former sending $x\in B_i$ to $i$ and the latter sending $i$ to $i$, but $B$ is not countable.