User:Dfeuer/Minimal Enclosing Complete Lattice is Order Completion

Theorem
Let $(T,\le)$ be a complete lattice.

Let $S \subseteq T$.

Suppose that for any $U$ such that $S \subseteq U \subsetneqq T$,


 * $(U, \le)$ is not complete.

Then $T$ is an order completion of $S$.

Proof
The identity mapping on $T$ restricted to $U$ is trivially order-preserving.