Does deleting a subset of an infinite set that has strictly smaller cardinality leave the cardinality of the infinite set unchanged?

Question

Is it a theorem of ZFC that if

1. $X \subseteq Y$,
2. $|X| < |Y|$,
3. $Y$ is infinite,

then

• $|Y \setminus X|=|Y|$?

Answer 1

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 $|Y \setminus X| = |Y|$.