User:Dfeuer/Empty Class is Supercomplete
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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.
$\blacksquare$