I've found some time to read a little more on set theory, and I've come across the following question.
Suppose I have four sets $X$, $Y$, $Z$, and $W$ such that $Y\subseteq W$ and $Z\subseteq X$. Suppose also that $X\cup Y\sim Y$, where by $\sim$ I mean that the two sets $X\cup Y$ and $Y$ are equinumerous. How can I show that $Z\cup W\sim W$?
I thought the Bernstein-Schroeder theorem might be applicable. The identity function maps $W$ into $Z\cup W$ injectively, so I figured it suffices to show that there is an injection from $Z\cup W$ into $W$. From $X\cup Y\sim Y$, there is an injection $f\colon X\cup Y\to Y$, and thus $f|_X$ is an injection from $X$ into $Y$. Since $Z\subseteq X$ there is an injection from $Z$ to $X$, and likewise from $Y$ into $W$. Composing all these would give an injection from $Z$ into $W$. Those were my thoughts, but I don't think I can use them to show that $Z\cup W$ maps injectively into $W$. There must be a better way. Thanks for any help.