# Category:Axioms/Basic Universe Axioms

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This category contains axioms related to Basic Universe Axioms.

A **basic universe** $V$ is a universal class which satisfies the following axioms:

### $\text A 1$: Axiom of Transitivity

- $V$ is a transitive class.

### $\text A 2$: Axiom of Swelledness

- $V$ is a swelled class.

### $\text A 3$: Axiom of the Empty Set

The empty class $\O$ is a set, that is:

- $\O \in V$

### $\text A 4$: Axiom of Pairing

Let $a$ and $b$ be sets.

Then the class $\set {a, b}$ is likewise a set.

### $\text A 5$: Axiom of Unions

Let $x$ be a set (of sets).

Then its union $\bigcup x$ is also a set.

### $\text A 6$: Axiom of Powers

Let $x$ be a set.

## Subcategories

This category has only the following subcategory.

### Z

## Pages in category "Axioms/Basic Universe Axioms"

The following 11 pages are in this category, out of 11 total.