Superset Relation is Compatible with Subset Product

Theorem

Let $\struct {S, \circ}$ be a magma.

Let $\circ_\PP$ be the subset product on $\powerset S$, the power set of $S$.

Then the superset relation $\supseteq$ is compatible with $\circ_\PP$.

Proof

By Subset Relation is Compatible with Subset Product, the subset relation $\subseteq$ is compatible with $\circ_\PP$.

From Inverse of Subset Relation is Superset, the inverse of $\subseteq$ is $\supseteq$.

The result follows from Inverse of Relation Compatible with Operation is Compatible.

$\blacksquare$