Cowen's Theorem/Lemma 5

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:
 * $x \subseteq y \implies M_y \subseteq M_x \cup \paren {\powerset y \setminus \powerset x}$

Proof
Let us recall the definition of $x$-special $g$.


 * $S$ is $x$-special ( $g$)



Let $x \subseteq y$.

Let $S = M_x \cup \paren {\powerset y \setminus \powerset x}$

First we show that $S$ is $y$-special $g$.

We take the criteria one by one:


 * $(1): \quad \O \in S$


 * $(2): \quad S$ is closed under $g$ relative to $x$


 * $(3): \quad S$ is closed under chain unions