Equivalence of Definitions of Semiring of Sets/Definition 1 implies Definition 2

Theorem
Let $\SS$ be a system of sets satisfying the semiring of sets axioms:

Then $\SS$ satisfies the semiring of sets axioms:

Proof
Let $\SS$ be a system of sets satisfying the axioms:

It remains to be shown that $\SS$ satisfies the axiom

Let $A, B \in \SS$.

Let $A_1 = A \cap B$.

By axiom $(2)$:
 * $A_1 \in \SS$

From Intersection is Subset:
 * $A_1 \subseteq A$

By axiom $(3)$:
 * $\exists$ a finite sequence of pairwise disjoint sets $A_2, A_3, \ldots, A_n \in \SS : \ds A = \bigcup_{k \mathop = 1}^n A_k$

Then:

As $A$ and $B$ were arbitrary, then $\SS$ satisfies axiom $(3')$

The result follows