Consider the statement



> 
> 
> > 
> > For any infinite set $X$ there is an injection $\varphi$ from $(X\times\{0\}) \cup (X\times\{1\})$ into $X$.
> > 
> > 
> > 
> 
> 
> 


Does this imply the ${\sf AC}$?



Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?
It's way over my head, but it seems that this question was answered in the negative by Gershon Sageev, An independence result concerning the axiom of choice, Ann. Math. Logic 8 (1975), 1-184.


