Characterization of Compact Element in Complete Lattice/Statement 3 implies Statement 2
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Theorem
Let $L = \struct{S, \preceq}$ be a complete lattice.
Let $a \in S$ satisfy:
- $\forall A \subseteq S : a \preceq \sup A \implies \exists F \subseteq A : F$ is finite $: a \preceq \sup F$
Then:
- $\forall I \subseteq S : I$ is an ideal $: a \preceq \sup I \implies a \in I$
Proof
Let:
- $I \subseteq S : I$ is an ideal $: a \preceq \sup I$
We have by hypothesis:
- $\exists F \subseteq I : F$ is finite $: a \preceq \sup F$
By Join Semilattice Ideal Axiom $\paren{\text {JSI} 2 }$: Subsemilattice of Join Semilattice:
- $\sup F \in I$
By Join Semilattice Ideal Axiom $\paren{\text {JSI} 1 }$: Lower Section of Join Semilattice:
- $a \in I$
Since $I$ was arbitrary:
- $\forall I \subseteq S : I$ is an ideal $: a \preceq \sup I \implies a \in I$
$\blacksquare$