Equivalence of Definitions of Semiring of Sets

Theorem
The definitions of a semiring of sets are equivalent.

That is, a collection $\mathcal S$ of subsets of a set $X$ is a semiring (of sets) if and only if:
 * $(1):\quad \varnothing \in \mathcal S$
 * $(2):\quad A, B \in \mathcal S \implies A \cap B \in \mathcal S$; i.e., $\mathcal S$ is $\cap$-stable
 * $(3):\quad$ If $A, A_1 \in \mathcal S$ such that $A_1 \subseteq A$, then there exists a finite sequence $A_2, A_3, \ldots, A_n \in \mathcal S$ such that:
 * $(3a):\quad \displaystyle A = \bigcup_{k=1}^n A_k$
 * $(3b):\quad$ The $A_k$ are pairwise disjoint

We prove that criterion $(3)$ can be replaced by:
 * $(3'):\quad$ If $A, B \in \mathcal S$, then there exist finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \mathcal S$ such that $\displaystyle A \setminus B = \bigcup_{k=1}^n A_k$.

$(3)$ implies $(3')$
Let $X$ be a set, and let $\mathcal S$ be a collection of subsets of $X$.

Suppose that for all $A, A_1 \in \mathcal S$ such that $A_1 \subseteq A$, there exists a finite sequence of sets $A_2, A_3, \ldots, A_n \in \mathcal S$ such that:
 * $A_1, A_2, \ldots, A_n$ are pairwise disjoint
 * $\displaystyle A = \bigcup_{k=1}^n A_k$

Let $B \in \mathcal S$, and let $A_1 = A \cap B$.

It follows that $A_1 \in \mathcal S$, by definition.

Also, $A_1 \subseteq A$ by Intersection Subset.

Then:

as desired.

$(3')$ implies $(3)$
Now suppose that for all $A, B \in \mathcal S$, there exists a finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \mathcal S$ such that $\displaystyle A \setminus B = \bigcup_{k=1}^n A_k$.

Then $B$ is disjoint with each of the sets $A_k$.

Let $B \subseteq A$. Then:

as desired.