Cowen's Theorem/Lemma 7

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:
 * $M$ is closed under $g$ relative to $x$.

Proof
Let $x \in M$.

Then $x \in M_x$.

Because $x \subseteq x \cup \map g x$, we have from Lemma $3$:
 * $M_x \subseteq M_{x \cup \map g x}$

Hence:
 * $x \in M_{x \cup \map g x}$

Also:
 * $\map g x \subseteq x \cup \map g x$

and so:
 * $\map g x \in M_{x \cup \map g x}$

Also, from Lemma $6$:
 * $M_{x \cup \map g x} \subseteq M$

Hence:
 * $\map g x \in M$

and the result follows.