Cowen's Theorem/Lemma 2

Lemma for Cowen's Theorem
Let $g$ be a progressing mapping.

Let $x$ be a set.

Let $\powerset x$ denote the power set of $x$.

Let $M_x$ denote the intersection of the $x$-special subsets of $\powerset x$ $g$.

Let $M$ be the class of all $x$ such that $x \in M_x$.

We have that:
 * $\O \in M$

Proof
By Lemma $1$:
 * $\powerset \O$ is $\O$-special $g$.

By Power Set of Empty Set:
 * $\powerset \O = \set \O$

Hence:
 * $\set \O$ is $\O$-special $g$.

The only other subset of $\set \O$ is $\O$, which is by definition empty.

So $\set \O$ is the only $\O$-special subset of $\powerset \O$ $g$.

Hence:
 * $M_\O = \set \O$

Because $\O \in \set \O$, we have:
 * $\O \in M_\O$

and so:
 * $\O \in M$