Reduction from Clique to something else
$X = E$, so it remains to define the collection $A$. For each vertex $v_i$ in $V$ create $A_i$ as following: for each edge $x_t$ incident to the vertex $v_i$, add $x_t$ to $A_i$. Thus the edge $x_t=(v_i, v_j) \in E \Leftrightarrow A_i \cap A_j = \{x_t\}$, and every pair $A_i, A_j$ has at most one common element. So, if there are $k$ different elements in $A$ such that every two $A_i, A_j$ of them has non-empty intersection which is exactly the edge between vertices $v_i$ and $v_j$, then these $k$ vertices are pairwise connected meaning that they form a clique.