Complete Lattice is Bounded

Theorem

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.

Then

$L$ is bounded.

Proof

By definition of complete lattice:

$\O$ admits a supremum and an infimum.

By Infimum of Empty Set is Greatest Element:

$\forall x \in S: x \preceq \inf \O$

Thus by definition:

$L$ is bounded above.

By Supremum of Empty Set is Smallest Element:

$\forall x \in S: \sup \O \preceq x$

Thus by definition:

$L$ is bounded below.

Hence $L$ is bounded.

$\blacksquare$

Sources

• Mizar article YELLOW_0:condreg 3