If Ideal and Filter are Disjoint then There Exists Prime Ideal Including Ideal and Disjoint from Filter

== Theorem ==
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive lattice.
Let $I$ be an ideal in $L$.
Let $F$ be a filter on $L$ such that
:$I \cap F = \O$
Then there exists a prime ideal $P$ in $L$:
$I \subseteq P$ and $P \cap F = \O$
== Proof ==
Define $X := \set {P \in \map {\operatorname {Ids} } L: I \subseteq P \land P \cap F = \O}$
where $\map {\operatorname {Ids} } L$ denotes set of all ideals in $L$.
By Set is Subset of Itself:
:$I \in X$
We will prove that:
:$\forall Z: Z \ne \O \land Z \subseteq X \land \paren {\forall Y_1, Y_2 \in Z: Y_1 \subseteq Y_2 \lor Y_2 \subseteq Y_1} \implies \bigcup Z \in X$
Let $Z$ such that:
:$Z \ne \O$ and
:$Z \subseteq X$ and
:$\forall Y_1, Y_2 \in Z: Y_1 \subseteq Y_2 \lor Y_2 \subseteq Y_1$
By definition of $X$:
:$Z$ is set of subsets of $S$.
By definition of $X$:
:$\forall Y \in Z: Y$ is a lower section.
Then by Every Element is Lower implies Union is Lower:
:$J := \bigcup Z$ is lower section.
By definition of non-empty set:
:$\exists Y: Y \in Z$
By definition of subset:
:$Y \in X$
By definition of $X$:
:$I \subseteq Y$
By Set is Subset of Union/General Result:
:$Y \subseteq J$
By Subset Relation is Transitive:
:$I \subseteq J$
We will prove that:
:$\forall A, B \in Z: \exists C \in Z: A \cup B \subseteq C$
Let $A, B \in Z$.
By assumption:
:$A \subseteq B$ or $B \subseteq A$
By Union with Superset is Superset:
:$A \cup B = B$ or $A \cup B = A$
Thus by Set is Subset of Itself:
:$\exists C \in Z: A \cup B \subseteq C$
By definition of $X$:
:$\forall Y \in Z: Y$ is directed.
By Every Element is Directed and Every Two Elements are Included in Third Element implies Union is Directed:
:$J$ is directed.
By definition of ideal in ordered set:
:$J$ is an ideal in $L$.
We will prove that
:$J \cap F = \O$
Let $x \in J$.
By definition of union:
:$\exists A \in Z: x \in A$
By definition of subset:
:$A \in X$
By definition of $X$:
:$A \cap F = \O$
Thus by definitions of intersection and empty set:
:$x \notin F$
Thus by definition of $X$:
:$\bigcup Z \in X$
By Zorn's Lemma:
:there exists $Y \in X$: $Y$ is maximal set of $X$. 
Then by definition of $X$:
:$Y \in \map {\operatorname {Ids} } L$ and $I \subseteq Y$ and $Y \cap F = \O$
We will prove that:
:$Y$ is a prime ideal.
Let $x, y \in S$ such that:
:$x \wedge y \in Y$
:
:$x \notin Y$ and $y \notin Y$
Define $P_y = \map {\operatorname{finsups} } {Y \cap \set y}^\preceq$
By Finite Suprema Set and Lower Closure is Smallest Ideal:
:$Y \cup \set y \subseteq P_y$
By Set is Subset of Union:
:$Y \subseteq Y \cup \set y$
By Subset Relation is Transitive:
:$Y \subseteq P_y$
By Subset Relation is Transitive:
:$I \subseteq P_y$
By definition of singleton:
:$y \in \set y$
By definition of union:
:$y \in Y \cup \set y$
By definition of subset:
:$y \in P_y$
We will prove that
:$P_y \cap F \ne \O$
 
:$P_y \cap F = \O$
By definition of $X$:
:$P_y \in X$
By definition of minimal set:
:$Y = P_y$
A contradiction between $y \notin Y$ and $y \in P_y$
By definitions of non-empty set and intersection:
:$\exists v: v \in P_y \land v \in F$
Define $P_x = \map {\operatorname{finsups} } {Y \cap \set x}^\preceq$
Analogically:
:$\exists u: u \in P_x \land u \in F$
By Finite Subset Bounds Element of Finite Suprema Set and Lower Closure:
:$\exists u' \in Y: u \preceq u' \vee \sup \set x$
By Finite Subset Bounds Element of Finite Suprema Set and Lower Closure:
:$\exists v' \in Y: v \preceq v' \vee \sup \set y$
By Join is Associative:
:$\paren {v' \vee u'} \vee x = v' \vee \paren {u' \vee x}$
By Join Succeeds Operands: 
:$\paren {v' \vee u'} \vee x \succeq u' \vee x$
By Supremum of Singleton:
:$\sup \set x = x$
By definition of transitivity:
:$u \preceq v' \vee u' \vee x$
By definition of upper section:
:$u' \vee v' \vee x \in F$
Analogically:
:$u' \vee v' \vee y \in F$
By Filtered in Meet Semilattice:
:$\paren {u' \vee v' \vee x} \wedge \paren {u' \vee v' \vee y} \in F$
By definition of distributive lattice:
:$\paren {u' \vee v'} \vee \paren {x \wedge y} \in F$
By Directed in Join Semilattice:
:$u' \vee v' \in Y$
By Directed in Join Semilattice:
:$\paren {u' \vee v'} \vee \paren {x \wedge y} \in Y$
This contradicts $Y \cap F = \O$
== Sources ==
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Category:Prime Ideals (Order Theory)