Way Below Relation is Antisymmetric
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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