Preceding implies Way Below Closure is Subset of Way Below Closure
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Theorem
Let $\struct {S, \preceq}$ be an ordered set.
Let $x, y \in S$ such that
- $x \preceq y$
Then $x^\ll \subseteq y^\ll$
where $x^\ll$ denotes the way below closure of $x$.
Proof
Let $z \in x^\ll$.
By definition of way below closure:
- $z \ll x$
By Preceding and Way Below implies Way Below and definition of reflexivity:
- $z \ll y$
Thus by definition of way below closure:
- $z \in y^\ll$
$\blacksquare$
Sources
- Mizar article WAYBEL_3:12