Axioms of Set theory

A very good video explaining the axioms of Set theory. https://youtu.be/zcvsyL7GtH4

Following are the 9 axioms of Set Theory.

1. Axiom of extensionality: Two sets are equal (are the same set) if they have the same elements.
2. Axiom of regularity (also called the Axiom of foundation): If element of a set X is another set Y, then X and Y are disjoint. This means Y cannot contain any elements that X contains, including Y itself. Hence, this prevents Y from containing itself.
3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension): In order to avoid paradoxes, if you are building a set X by describing some property of its elements, then you need to specify another set Y from where you need to pick the elements of X in question. (Note: Wikipedia says that this restriction is necessary to avoid Russell's paradox, but I do not see how, since the Paradox can be solved simply by applying the axiom of Regularity- Set that contains itself does not exist in the first place. Russell's paradox questions the existence of set of all sets that do not contain themselves.)
4. Axiom of pairing: If x and y are sets, then there exists a set which contains x and y as elements.
5. Axiom of union: The union over the elements of a set exists
6. Axiom schema of replacement: When a function f acts on a set X, the image or the range of X is also a set.
7. Axiom of infinity: There exist a set of infinite size.
8. Axiom of power set: Power set exists for every set.
9. Axiom of choice: Given a set X that contains other non empty sets $Y_1, Y_2, Y_3$ etc. within it. Then there exists a function f from X to $Y_1 \cup Y_2 \cup Y_3 \cup...$ such that $f(Y_n) \in Y_n$.