Singleton of Power Set less Empty Set is Minimal Subset
Jump to navigation
Jump to search
Theorem
Let $S$ be a set which is non-empty.
Let $\CC = \powerset S \setminus \O$, that is, the power set of $S$ without the empty set.
Let $x \in S$.
Then $\set x$ is a minimal element of the ordered structure $\struct {\CC, \subseteq}$.
Proof
Let $y \in \CC$ such that $y \subseteq \set x$.
We have that $\O \notin \CC$.
Therefore:
- $\exists z \in S: z \in y$
But as $y \subseteq \set x$ it follows that:
- $z \in \set x$
and so by definition of singleton:
- $z = x$
and so:
- $y = \set x$
and so:
- $y = x$
Thus, by definition, $\set x$ is a minimal element of $\struct {\CC, \subseteq}$.
$\blacksquare$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order