We know the [Shroeder-Bernstein](http://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem) (SB) theorem can be proved in ZF, while the [Dual Schroeder-Bernstein](https://mathoverflow.net/questions/38771/dual-schroeder-bernstein-theorem) (DSB) can be proved in ZF+AC but not in ZF. Define as ISB the property that whenever there are both an injection and a surjection $X \to Y$, then there must be a bijection too. Trivially ZF+DSB $\implies$ ZF+ISB (because an injection $X\to Y$ allows to define a surjection $Y\to X$, without using AC). Is the converse true? I'm assuming here that ZF $\nRightarrow$ ISB, but I don't know a proof of that either (UPDATE: this is the case, as pointed out by Asaf Karagila in a comment).



Are two forms of the Dual Schroeder-Bernstein property equivalent?
What you call $\sf ISB$, is better known as $\sf WPP$ (Weak Partition Principle) and can be formulated as "*If there is a surjection from $X$ onto $Y$, then $X$ cannot have a strictly smaller cardinality than $Y$*" (alternatively, $|X|\leq^\*|Y|\leq|X|\rightarrow |X|=|Y|$ or $|X|\leq^\*|X|\rightarrow |X|\nless|Y|$).


In their paper Banaschewski and Moore write that this is still open, and Higasikawa makes no mention of a possible answer in his paper from 1995. I am unaware of any new papers since then that have dealt with these topics, meaning that this problem is *probably* still open.




---


*Bibliography.*


1. Bernhard Banaschewski, Gregory H. Moore, **[The dual Cantor-Bernstein theorem and the partition principle](http://dx.doi.org/10.1305/ndjfl/1093635502)**, *Notre Dame J. Formal Logic* **31 (3)**, (1990), 375–381.
2. Masasi Higasikawa, **[Partition principles and infinite sums of cardinal numbers](http://projecteuclid.org/euclid.ndjfl/1040149358).** *Notre Dame J. Formal Logic* **36** (1995), no. 3, 425–434.


Additional reading:


3. [A question about the Axiom of Choice](https://mathoverflow.net/questions/135945/a-question-about-the-axiom-of-choice)


