Does deleting a subset of an infinite set that has strictly smaller cardinality leave the cardinality of the infinite set unchanged? Is it a theorem of ZFC that if


*

*$X \subseteq Y$,

*$|X| < |Y|$,

*$Y$ is infinite,


then


*

*$|Y \setminus X|=|Y|$?


Yes it is, it follows from $\kappa + \lambda = \max(\kappa, \lambda)$ if at least one of the cardinals in question is infinite.
So suppose $|Y \setminus X| < |Y|$, than as $Y$ is infinte either $X$ or $|Y \setminus X|$ is infinite, so 
$$ |Y| = |Y \setminus X| + |X|  = \max(|X|, |Y \setminus X|) < |Y| $$
contradiction. Hence $|Y \setminus X| = |Y|$.
