Partition of a set with size conditions

High school · ●●●○○ 50 pts

Let

N

be the number of ordered pairs of nonempty sets

\mathcal{A}

and

\mathcal{B}

such that

\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}

\mathcal{A} \cap \mathcal{B} = \emptyset

, the number of elements of

\mathcal{A}

is not an element of

\mathcal{A}

, and the number of elements of

\mathcal{B}

is not an element of

\mathcal{B}

. Find

N