User:Dfeuer/Empty Class is Supercomplete

Theorem
The empty class is supercomplete.

That is, the empty class is transitive and swelled.

Proof
In order to be transitive, $\varnothing$ must satisfy:


 * $\forall x: x \in \varnothing \implies x \subseteq \varnothing$.

By the definition of the empty class:


 * $\forall x: \lnot (x \in \varnothing)$.

Thus $\varnothing$ is vacuously transitive.

In order to be swelled, $\varnothing$ must satisfy:


 * $\forall x: \forall y: (x \in \varnothing \land y \subseteq x \implies y \in \varnothing)$

Again, this is vacuously true.