Image of Proper Subset under Progressing Mapping on Minimally Closed Class

Theorem

Let $N$ be a class which is closed under a progressing mapping $g$.

Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$.

Then:

$x \subset y \implies \map g x \subseteq y$

Proof

From Minimally Closed Class under Progressing Mapping induces Nest, we have that $N$ is a nest in which:

$\forall x, y \in N: \map g x \subseteq y \lor y \subseteq x$

Thus the corollary 1 of the Sandwich Principle applies directly.

$\blacksquare$