Bottom in Ideal
Let $I$ be a ideal in $S$.
Then $\bot \in I$
where $\bot$ denotes the smallest element of $S$.
By definition of ideal in ordered set:
By definition of non-empty set:
- $\exists x: x \in I$
By definition of smallest element:
- $\bot \preceq x$
Thus by definition of lower set:
- $\bot \in I$
- Mizar article WAYBEL_4:21