China 1999

[Corei13]

Let, $A = \{1,2, \cdots n \}$
$\Im = \{(A_1,A_2,\cdots, A_n) | |A_i| \geq 2, |A_i \cap A_j| \leq 1 \text{ for } i\neq j \}$
And,
\[ \forall \{i,j\}\subseteq A, \text{ } \forall (A_1,A_2,\cdots, A_n) \in \Im, \text{ }\exists u, 1 \leq u \leq n \text{ such that } \{i,j\} \subseteq A_u, \text{ }\{i,j\} \not\subseteq A_v, \text{ }\forall v\neq u, 1\leq v \leq n \]
[ Here $\forall k$ means For all k and $\exists k$ means There exists a k ]
Prove that, $\forall (A_1,A_2,\cdots, A_n) \in \Im$, $|A_i \cap A_j| = 1$ for all $1 \leq i < j \leq n$