Axiom:Axioms of Class Existence

Axiom

Let the ordered pair notation be understood to denote:

$\tuple {X, Y} := \set {\set X, \set {X, Y}}$

The following are the axioms of class existence:

$(\text B 1)$	$:$	$\ds \exists X: \forall u, v: \tuple {u, v} \in X \iff u \in v $	$\in$-relation
$(\text B 2)$	$:$	$\ds \forall X, Y: \exists Z: \forall u: u \in Z \iff u \in X \land u \in Y $	intersection
$(\text B 3)$	$:$	$\ds \forall X: \exists Z: \forall u: u \in Z \iff u \notin X $	complement
$(\text B 4)$	$:$	$\ds \forall X: \exists Z: \forall u: u \in Z \iff \exists v: \tuple {u, v} \in X $	domain
$(\text B 5)$	$:$	$\ds \forall X: \exists Z: \forall u, v: \tuple {u, v} \in Z \iff u \in X $
$(\text B 6)$	$:$	$\ds \forall X: \exists Z: \forall u, v, w: \tuple {\tuple {u, v}, w} \in Z \iff \tuple {\tuple {v, w}, u} \in X $
$(\text B 7)$	$:$	$\ds \forall X: \exists Z: \forall u, v, w: \tuple {\tuple {u, v}, w} \in Z \iff \tuple {\tuple {u, w}, v} \in X $

\((\text B 1)\)	$:$	\(\ds \exists X: \forall u, v: \tuple {u, v} \in X \iff u \in v \)	$\in$-relation
\((\text B 2)\)	$:$	\(\ds \forall X, Y: \exists Z: \forall u: u \in Z \iff u \in X \land u \in Y \)	intersection
\((\text B 3)\)	$:$	\(\ds \forall X: \exists Z: \forall u: u \in Z \iff u \notin X \)	complement
\((\text B 4)\)	$:$	\(\ds \forall X: \exists Z: \forall u: u \in Z \iff \exists v: \tuple {u, v} \in X \)	domain
\((\text B 5)\)	$:$	\(\ds \forall X: \exists Z: \forall u, v: \tuple {u, v} \in Z \iff u \in X \)
\((\text B 6)\)	$:$	\(\ds \forall X: \exists Z: \forall u, v, w: \tuple {\tuple {u, v}, w} \in Z \iff \tuple {\tuple {v, w}, u} \in X \)
\((\text B 7)\)	$:$	\(\ds \forall X: \exists Z: \forall u, v, w: \tuple {\tuple {u, v}, w} \in Z \iff \tuple {\tuple {u, w}, v} \in X \)