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:

$\Bbb U = \set {x: x = x}$

From this, it follows immediately that:

$\forall x: \paren {x \in \Bbb U \iff x = x}$

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

Thus the asserted statement is also tautologous.

$\blacksquare$

Sources

• 1963: Willard Van Orman Quine: Set Theory and Its Logic: $\S 6.8$