$\DeclareMathOperator{\card}{card}$ Suppose $\card(C)= \card(D)$, $D \subseteq C$, and $\card(C\setminus D)$ is infinite.
I want to show that $\card(C\setminus D) = \card(C)$.
This is easy enough for me to see in some specific cases. For example, $\card(\Bbb N=\card(\Bbb Q)$, $\Bbb N\subseteq\Bbb Q$, and $\card(\Bbb Q\setminus\Bbb N) = \card(\Bbb Q)$.
But because subtraction of two equal, infinite cardinals is not well defined, I am unsure of how to prove the general case. What would be a good method to go about it?