Way Below Relation is Antisymmetric

Theorem

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

Let $x, y \in S$ such that

$x \ll y$ and $y \ll x$

Then

$x = y$

Proof

By Way Below implies Preceding:

$x \preceq y$ and $y \preceq x$

Thus by definition of antisymmetry:

$x = y$

$\blacksquare$

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_3:6