$\lambda$ (Dynkin) system equivalent definitions, proof?

Let $A_1 \subset A_2 \subset \cdots$ be as described in (F3). Let $B_1=A_1$ and $B_i := A_i \setminus A_{i-1}$ for $i \ge 2$. Apply (S3) to $B_1,B_2,\ldots$ and note that $\bigcup_i A_i = \bigcup_i B_i$.

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(S1) = (F1)

(S2): Apply (F2) with $B=\Omega$.

(S3): Let $A_1,A_2,\ldots$ be as described in (S3). Let $B_n:=\bigcup_{i=1}^n A_i$. Apply (F3) to $B_1,B_2,\ldots$ and note that $\bigcup_n A_n = \bigcup_n B_n$.