Preceding and Way Below implies Way Below

Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Let $u, x, y, z \in S$ such that

$u \preceq x \ll y \preceq z$

where $\ll$ denotes the way below relation.

Then

$u \ll z$

Proof

Let $D$ be a directed subset of $S$ such that

$D$ admits a supremum

and

$z \preceq \sup D$

By definition of transitivity:

$y \preceq \sup D$

By definition of way below relation:

$\exists d \in D: x \preceq d$

Thus by definition of transitivity:

$\exists d \in D: u \preceq d$

Thus by definition of way below relation:

$u \ll z$

$\blacksquare$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_3:2