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

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 $A, B \in \SS : B \subseteq A$.

By axiom $(3)$:
 * $\exists$ a finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \SS : \ds A \setminus B = \bigcup_{k \mathop = 1}^n A_k$.

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

Then:

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

The result follows