# Category:Axioms/Von Neumann-Bernays-Gödel Axioms

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This category contains axioms related to Von Neumann-Bernays-Gödel Axioms.

### The Axiom of Extension

Let $A$ and $B$ be classes.

Then:

- $\forall x: \paren {x \in A \iff x \in B} \iff A = B$

### The Axiom of Specification

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 needs considerable tedious hard slog to complete it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Subcategories

This category has the following 7 subcategories, out of 7 total.

### A

- Axioms/Axiom of Empty Set (8 P)
- Axioms/Axiom of Extension (10 P)
- Axioms/Axiom of Infinity (9 P)
- Axioms/Axiom of Pairing (8 P)
- Axioms/Axiom of Powers (7 P)
- Axioms/Axiom of Unions (8 P)

## Pages in category "Axioms/Von Neumann-Bernays-Gödel Axioms"

The following 17 pages are in this category, out of 17 total.