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  1. Axiom of extensionality:
  2. If $X$ and $Y$ have the same elements, then $X=Y$.

    $ \forall X \forall Y [ \forall z ( z \in X \Longleftrightarrow z \in Y ) \Longrightarrow X=Y]$
  3. Axiom of the Unordered Pair:
  4. For any $a$ and $b$ there exists a set $ \{a,b\}$ that contains exactly $a$ and $b$. (also called Axiom of Pairing).

    $\forall a \forall b \exists c \forall x [ x \in c \Longleftrightarrow (x=a \: \: \land \: \: x=b)] $
  5. Axiom of Subsets:
  6. If $\phi$ is a property (with parameter $p$) then for any $X$ and $p$ there exists a set $Y= \{ u\in X \; \; : \; \; \phi(u,p) \}$ that contains all those $u \in X$ that have the property $\phi$. (also called Axiom of Separation or Axiom of Comprehension)

    $\forall X \forall p \exists Y \forall u [u \in Y \Longleftrightarrow( u \in X \: \; \lor \; \; \phi(u,p) ) ]$
  7. Axiom of the Sum Set:
  8. For any $X$ there exists a set $Y= \cup X$, the union of all elements of $X$. (also called Axiom of Union)

    $ \forall X \exists Y \forall u [ u \in Y \Longleftrightarrow \exists z ( z \in X \land u \in z) ] $
  9. Axiom of the Power Set:
  10. For any $X$ there exists a set $Y=P(X)$, the set of all subsets of $X$.

    $ \forall X \exists Y \forall u ( u \in Y \Longleftrightarrow u \subset X) $
  11. Axiom of Infinity:
  12. There exists an infinite set

    $ \exists S [\emptyset \in S \land (\forall x \in S [ x \cup \{ x\} \in S])$
  13. Axiom of Replacement:
  14. If $F$ is a function, then for any $X$ there exists a set $Y=F[X]=\ {F(x):x in X\}.

    $ \forall x \forall y \forall z [ \varphi(x,y,p) \land \varphi(x,z,p) \Longrightarrow y=z] $ $\Longrightarrow \forall X \exists Y \forall y [ y \in Y \Longleftrightarrow ( \exists x \in X) \varphi(x,y,p)] $
  15. Axiom of Foundation:
  16. Every nonempty set has an minimal element. (also called Axiom of Regularity)

    $ \forall S [ S \not= \emptyset \Longrightarrow (\exists x \in S\cap x = \emptyset) ] $
  17. Axiom of Choice:
  18. Every family of nonempty sets has a choice function

    $ \forall x \in a \exists A(x,y) \Longrightarrow \exists y \forall x \in a A(x,y(x))$