Stable matching with constraints
I assume you want to start your proof with "suppose there is a pairing where a good man $m_1$ is matched with a bad woman $w_1$".

This indeed means that there exists a good woman $w_2$ matched with a bad man $m_2$ (by the pigeonhole principle). However, the contradiction here does not follow from the fact that $w_2$ prefers $m_1$ but is actually matched to $m_2$ (it is possible for a woman to not be matched with her top ranked man).

The contradiction follows from the fact that a matching which contains $(m_1,w_1),(m_2,w_2)$ is not stable, since $m_1$ prefers $w_2$ and $w_2$ prefers $m_1$.