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
Suppose that
 * $\forall x: ( x \subseteq A \implies x \in A )$.

Consider $U \setminus A$.

Suppose for the sake of contradiction that $U \setminus A \ne \varnothing$.

Then by Axiom of Foundation (Strong Form), we have that:


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

But:

Thus $x \setminus A = \varnothing$, so

$x \subseteq A$.

By hypothesis, then, $x\in A$, contradicting the fact that $x \notin A$.

Therefore we can conclude that
 * $( U \setminus A ) = \varnothing$, so
 * $U \subseteq A$.

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