Epsilon Induction

Theorem
Let $A$ be a class.

Let $U$ denote the universe.


 * $\forall x: ( x \subseteq A \implies x \in A ) \implies A = U$

Proof
Consider $U \setminus A$.

If $U \setminus A \ne \varnothing$, then by Axiom of Foundation (Strong Form), we have that:


 * $\exists x \notin A: ( x \cap ( U \setminus A ) ) = \varnothing$

But:

So $( U \setminus A ) = \varnothing$ and $U \subseteq A$.

Furthermore, $A \subseteq U$, so $A = U$.