User:TurdProtector/sandbox

Equality
We define equality as follows:
 * $$ a = b : \stackrel{\mathrm{definition}}{\Longleftrightarrow} \ \forall x: \ x \in a \Leftrightarrow x \in b $$

It follows:
 * Reflexivity.
 * $$a = a$$


 * $$\forall x \ x \in a \Leftrightarrow x \in a$$, which always holds


 * Symmetry
 * $$a = b \Rightarrow b = a$$


 * Transitivity
 * $$a = b \land b = c \Rightarrow a = c $$

The axioms

 * Extensionality, two sets with the same elements are equal.
 * $$\forall x,y,z \ (z \in x \Leftrightarrow z \in y) \Rightarrow (x=y)$$


 * Separation, subsets exist
 * $$\forall y_1,p \ \exists y_2 \ \forall x\  x\in y_2 \Leftrightarrow

(p \wedge x\in y_1)$$ where p is any proposition
 * The empty set exists
 * $$\exists x \ \forall y\ y\not\in x$$


 * Union, the union of all members of a set is a set.
 * $$\forall x\ \exists  y\  \forall z \ z\in y \Leftrightarrow (\exists u \ z\in u \wedge u\in x)$$


 * Power sets exist
 * $$\forall x \ \exists y \ \forall z \ z\in y \Leftrightarrow

(\forall t \ t \in z \Rightarrow t \in x)$$ we denote this set y by P(x)
 * Infinity, an infinite set exists
 * $$\exists x \ (\empty \in x) \wedge ( \forall y \ y\in x \Rightarrow P(y)\in x)$$


 * Foundation, no set is a member of itself
 * $$\forall x \ x \ne \empty \Rightarrow (\exists y \in x \ y \cap x = \empty)$$