Fundamental Law of Universal Class

Theorem

 * $\forall x: x \in \Bbb U$

where:
 * $\Bbb U$ denotes the universal class
 * $x$ denotes a set.

Proof
From the definition of the universal class:


 * $\Bbb U = \left\{{x: x = x}\right\}$

From this, it follows immediately that:


 * $\forall x: \left({x \in \Bbb U \iff x = x}\right)$

From Equality is Reflexive, $x = x$ is a tautology.

Thus the asserted statement is also tautologous.

Source

 * : $\S 6.8$