I’ve not seen a version that exactly matches this description, but the following comes close. It considers only smaller sets, but it compensates by considering both sets of boys and sets of girls.
If $B$ and $G$ are the sets of boys and girls, respectively, for each $b \in B$ let $\overline G(b)$ be the set of girls whom $b$ does not like, and for each $g \in G$ let $\overline B(g)$ be the set of boys whom $g$ does not like. (It’s more convenient here to use these rather than $G(b)$ and $B(g)$, the sets of boys or girls liked by an individual, but the two are obviously interchangeable.) Then the boys and girls can be married off iff there do not exist $B_0 \subseteq B$ and $G_0 \subseteq G$ such that $|B_0| + |G_0| > n$, $G_0 \subseteq \bigcap_{b \in B_0} \overline G(b)$, and $B_0 \subseteq \bigcap_{g \in G_0} \overline B(g)$. Clearly if such $B_0$ and $G_0$ did exist, one of them would have to have cardinality at most $\lceil n/2 \rceil$.
Suppose that for each $B_1 \subseteq B$ and $G_1 \subseteq G$ of cardinality at most $\lceil n/2 \rceil$, $\left|\bigcap_{b \in B_1} \overline G(b) \right| < n - |B_1|$ and $\left|\bigcap_{g \in G_1} \overline B(b) \right| < n - |G_1|$. Suppose, to get a contradiction, that $B_0 \subseteq B$ and $G_0 \subseteq G$ are such that $|B_0| + |G_0| > n$, $G_0 \subseteq \bigcap_{b \in B_0} \overline G(b)$, and $B_0 \subseteq \bigcap_{g \in G_0} \overline B(g)$; without loss of generality $|B_0 \le \lceil n/2 \rceil$. But then $\left|\bigcap_{b \in B_0} \overline G(b) \right| < n - |B_0| < |G_0|$, so $G_0 \not\subseteq \bigcap_{b \in B_0} \overline G(b)$.
It only remains to check that the nonexistence of sets $B_0 \subseteq B$ and $G_0 \subseteq G$ such that $|B_0| + |G_0| > n$, $G_0 \subseteq \bigcap_{b \in B_0} \overline G(b)$, and $B_0 \subseteq \bigcap_{g \in G_0} \overline B(g)$ is equivalent to the usual formulation of the marriage condition, but this is pretty straightforward if one thinks in terms of the adjacency matrix of the obvious bipartite graph, as in this discussion.