ZFC axiom System of Set theory

This is the most common axiom system of set theory in mathematics, which is equivalent to the axiom system proposed by Kunen in 1980.
(ZF1) Extension axiom: A set is entirely determined by its elements. If two sets contain the same elements, they are equal.
(ZF2) An empty set has the axiom that there is a set of S, which has no elements.
(ZF3) unordered-to-axiom: that is, any given two sets X, Y, there is a third set Z, which makes w∈z when and only if W=x or w=y. What this axiom actually says is that given two sets of X and Y, we can find a set of a, whose members are exactly x and Y.
(ZF4) and set axiom: That is, to give a set of X, we can bring together elements of X to form a new set.
The exact definition: "For any set X, there is a set of Y, so that w∈y when and only if there is Z make z∈x and W∈z".
(ZF5) Power set axiom: That is, any set x,p (x) is also a set.
The exact definition: "For any set X, there is a set Y, so that z∈y when and only if all elements of Z are W,w∈x".
(ZF6) Infinite axiom: In other words, there is a set of X, which has an infinite number of elements.
The exact definition: "There is a collection where the empty set is its element, and x,x∪{x} is its element for any of its elements." "According to the description of the natural number in the system of the axiom of the skin, this is the existence of a set containing all the natural numbers."
(ZF7) Separation axiom Pattern: "The logical predicate P (z), defined for any set X and any element of x, exists with the set Y, so that z∈y when and only if Z∈x and P (z) are true".
(ZF8) Substitution axiom pattern: that is, for any function f (x), for any set T, when x belongs to T, F (x) has a definition (the only object in ZF is a set, so f (x) must be a set), there must be a set of s, so that for all x belongs to T, There is one element y in the set S, so Y=f (x). That is, when the defined field of a function defined by f (x) is in T, its value range is limited to S.
(ZF9) A regular axiom: Also called the fundamental axiom. All sets are Liangji sets. Indicates that the elements of a collection have minimal properties, for example, where x is not allowed.
The exact definition: "for any non-empty collection x,x at least one element y makes X∩y an empty set." ”
Note 1: All of the above is the content of the ZF Axiom system, plus the choice axiom constitutes the ZFC axiom system.
(AC) Selection axiom: For any set C existence in C as the definition of the field of the selection function g, so that each non-empty tuple of C x,g (x) ∈x.
Note 2: The empty set axiom can be derived from other axioms. It is generally believed that the ZF axiom system may not contain the empty set axiom.