Universal Class is Proper

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Theorem

Let $V$ denote the universal class.


Then $V$ is a proper class.


Proof 1

Aiming for a contradiction, suppose $V$ is small.

We have that:

$\operatorname {Ru} \subseteq V$

where $\operatorname {Ru}$ denotes the Russell class.

By Axiom of Subsets Equivalents, $\operatorname {Ru}$ is also small.


This contradicts Russell's Paradox.

$\blacksquare$


Proof 2

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Let $I_V$ be the identity mapping on $V$.

By Identity Mapping is Bijection it follows that $I_V$ is a bijection.


Therefore, by the Axiom of Limitation of Size, $V$ is proper.

$\blacksquare$


Proof 3

Aiming for a contradiction, suppose $V$ is small.

By Cantor's Theorem, there is no surjection from $V$ to $\powerset V$.

By definition of universal class:

$\powerset V \subseteq V$

By Injection from Subset to Superset, there exists an injection from $\powerset V$ to $V$.

By Injection has Surjective Left Inverse Mapping, there is a surjection from $V$ to $\powerset V$.

But this contradicts Cantor's Theorem.

Therefore $V$ is proper.

$\blacksquare$


Sources