Axiom:Axiom of Specification/Class Theory

Axiom
The axiom of specification is an axiom schema which can be formally stated as follows:

Let $\map \phi {A_1, A_2, \ldots, A_n, x}$ be a propositional function such that:
 * $A_1, A_2, \ldots, A_n$ are a finite number of free variables whose domain ranges over all classes
 * $x$ is a free variable whose domain ranges over all sets

Then the axiom of specification gives that:
 * $\forall A_1, A_2, \ldots, A_n: \exists B: \forall x: \paren {x \in B \iff \map \phi {A_1, A_2, \ldots, A_n, x} }$

where each of $B$ ranges over arbitrary classes.

This means that for any finite number $A_1, A_2, \ldots, A_n$ of subclasses of the universal class $V$, the class $B$ exists (or can be formed) of all sets $x \in V$ that satisfy the function $\map \phi {A_1, A_2, \ldots, A_n, x}$.

Allowing bound proper class variables to be hidden inputs for $\phi$ results in Morse-Kelley set theory, which is strictly stronger than Gödel-Bernays set theory.

Also see

 * Axiom:Axiom of Specification/Set Theory