Operand is Upper Bound of Way Below Closure

Theorem

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

Let $x \in S$.

Then

$x$ is upper bound for $x^\ll$

where $x^\ll$ denotes the way below closure of $x$.

Proof

Let $y \in x^\ll$

By definition of way below closure:

$y \ll x$

where $\ll$ denotes the way below relation.

Thus by Way Below implies Preceding:

$y \preceq x$

Thus by definition:

$x$ is upper bound for $x^\ll$

$\blacksquare$

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_3:9