Drake, Seven Axioms of the Algebra of Events On page 3 of Drake Fundamentals of Applied Probability he lists The Seven Axioms of the Algebra of Events.
$1. A \cup B = B \cup A \\
2. A \cup (B \cup C) = (A \cup B) \cup C \\
3. A\cap( B \cup C) = A \cap B \cup A \cap C \\
4. (A')' = A \\
5. (A\cap B)' = A' \cup B'\\
6. A\cap A' = \phi \\
7.A \cap U = A  $
I have two questions.
First while Drake states that his choice is not unique is there a reason that he selected these seven and is seven the minimum possible number of axioms with which the Algebra of events can be expressed.
Second he further states that any other relation in the algebra of events can be proved directly from these seven axioms with no additional information and in fact in the chapter problem section asks the student to do so. Does the author mean that all the other set relations in the Algebra of Events can be proved by direct manipulation of the seven axioms with out resorting to the method of $x \in A$
For example, proofs of $A \cup A' = U$ are given in this posting
Proof of union of a set and its complement is equivalent to a universe
but these proofs use the method of $x \in A$ and not direct manipulation of Axiomatic statements as required by Drake. How would I prove $A \cup A' = U$ or $A \cup A = A$ using only direct manipulation of Drake's Seven Axioms of the Algebra of Events?

remark
From 1-7 we deduce that $U$ is unique.  Suppose
$$A \cap U = A  \text{ for all }A\tag7$$
$$A \cap V = A  \text{ for all }A\tag{7'}$$
We claim $U = V$.
Proof. From $(7)$ we get $V \cap U = V$.
From $(7')$ we get $U \cap V = U$.
Then apply $(1)$ to get $V \cap U = U \cap V$, and conclude $U = V$.
