Consider the following problem.
Let $S=\{1,2,\dots,n\}$, and $A_{ij}\subseteq S$ for $i,j\in S$. Does there exist a strict subset $T\subset S$ such that for all $j\not\in T$, there exists $i\in T$ for which $T\subseteq A_{ij}$?
(For $T=S$, this is trivially true, so we disallow it.)
The problem is in NP: given a set $T$, it is easy to verify whether it satisfies the property. Is it also NP-complete, or does there exist a polytime algorithm?
I thought about reducing from SAT. So we start with constraints like $$(x_1\vee \neg x_2\vee x_4)\wedge(\neg x_3\vee \neg x_4)\wedge\dots.$$ Each $x_i$ can refer to $i\in T$. So we want to express statements like
$$(1\in T\vee 2\not\in T\vee 4\in T)\wedge(3\not\in T\vee 4\not\in T)\wedge\dots$$
How can we do this?
It will be less confusing if we allow ourselves to label the elements of $S$ by something else than natural numbers, so let's instead take $$ S=\{c_1,c_2,\ldots,c_k,x_1,\bar x_1,x_1,\bar x_2,\ldots,x_n,\bar x_n\} $$ with one element per clause in your SAT problem and two elements for each variable -- one will be set if the variable is true, the other if the variable is false. Splitting a variable in two like this is very common trick for reducing from SAT.
The elements $c_k$ will be in none of the $A_{ij}$s. Since by specification there must be some element missing from $T$, this will force all of the $c_i$s be absent from $T$.
Can you select values for $A_{ix_j}$ and $A_{i\bar x_j}$ that force $T$ to contain exactly one of $x_j$ and $\bar x_j$, and values for $A_{ic_j}$ that enforce each of the clauses in your incoming SAT problem?