A Question From Shiryaev I was studying about $\lambda$-systems from Shiryaev where I countered the following statement about equivalence of conditions for defining a $\lambda$-system.

I can check the equivalence of the second set of conditions given the first. However, going the other way, I am not able to show condition $(\lambda_{b})$ given conditions $(\lambda_{a}), (\lambda'_{b})$ and $(\lambda'_{c})$.
Can someone please provide a hint on how to prove this?

Hint: If $A,B \in \mathscr{L}$ and $A \subseteq B$, then can you show (In my notation, $B^c$ is the complement of $B$)
a) $B^c \in \mathscr{L}$
b)$B^c\bigcup A  \in \mathscr{L}$ (Since $A\subseteq B \iff A\cap B^c = \phi$).
c)$B^c\bigcup A = (B\setminus A)^c$ 
