The measured explosion of 1933

From the three rather spartan axioms of set-theoretic probability theory a whole world of results follow by proof of lemmas of increasing complexity. To help along the way we can now steal some of the basic findings of set theory.  I won't go into detail on them but take them as read.

1. $\exists \emptyset$
2. $\exists E$, the sample space
3. Set sizes can be finite, countably infinite and uncountably infinite
4. All subsets of the integers are at most countably infinite
5. The set of real numbers is uncountably infinite
6. The set of real numbers in the $\left[0,1\right]$ interval is also uncountably infinite
7. $A\cup A = A$
8. $A \cup \emptyset = A$
9. $A \cup E = E$
10. $A \cup B = B \cup A$
11. $A \cup B \cup C = A \cup (B \cup C) = (A \cup B) \cup C$
12. $A \cap \emptyset = A$
13. $A \cap A = A$
14. $A \cap E = E$
15. $A \cap B = B \cap A$
16. $A \cap B \cap C = A \cap (B \cap C) = (A \cap B) \cap C$
17. $(A^c)^c=A$
18. $\emptyset^c=E$
19. $S^c=\emptyset$
20. $A \cup A^c = S$
21. $A \cap A^c = \emptyset$