So if I'm given $A \subseteq^{*} B$ and $B \subseteq^{*} C$ then I must prove that $A \subseteq^{*} C$ also holds. So far my best attempt has been to try and prove that $A-C \subseteq (A-B) \cup (B-C)$. Surely the union of two finite sets is finite and any subset of a finite set must also be finite. Here is how my proof of that is going:
Let $x \in A-C$ then $x \in A \text{ and } x \notin C$
1-$\Longrightarrow (x \in A \wedge x \notin C) \wedge (x \in B \vee x \notin B)$
2-$\Longrightarrow ((x \in A \wedge x \notin C) \wedge x \in B) \vee ((x \in A \wedge x \notin C) \wedge x \notin B)$
3-$\Longrightarrow (x \in B \wedge x \notin C \wedge x \in A) \vee (x \in A \wedge x \notin B \wedge x \notin C)$
4-$\Longrightarrow (x \in B-C \wedge x \in A) \vee (x \in A-B \wedge x \notin C)$
5-$\Longrightarrow (x \in B-C) \vee (x \in A-B)$
6-$\Longrightarrow x \in(B-C) \cup (A-B)$
7-$\Longrightarrow x \in (A-B) \cup (B-C) $
I'm not entirely sure of my steps 1 and 5. On step one I use the law of excluded middle to intrduce statements, is this allowed when proving things about sets? . On step five I essentially eliminated parts of the statement, the justification being the logic law disjucntion elimination. Essentially, if both statements of an or lead to the same conclusion it is then safe to conclude that.
So I guess my question is a double one, is my method of proving that the described relaiton is transitive valid? and second is there a better method? Perhaps one where you don't have to prove that one set is a subset of the other (and thus don't have to rely on the assumptions that union and subsets of finite sets are finite, as obvious as that assumption might be).