Pow closed under binary intersection

2 files changed, 16 insertions, 8 deletions

diff --git a/library/set.tex b/library/set.tex
index fcd2642..2fd18ea 100644
--- a/library/set.tex
+++ b/library/set.tex
@@ -551,14 +551,6 @@ The $\operatorname{\textsf{cons}}$ operation is determined by the following axio
     Follows by set extensionality.
 \end{proof}
 
-\begin{proposition}\label{inter_subseteq_left}
-    $A\inter B\subseteq A$.
-\end{proposition}
-
-\begin{proposition}\label{inter_subseteq_right}
-    $A\inter B\subseteq B$.
-\end{proposition}
-
 \begin{proposition}\label{inter_emptyset}
     $A\inter\emptyset = \emptyset$.
 \end{proposition}
@@ -620,6 +612,11 @@ The $\operatorname{\textsf{cons}}$ operation is determined by the following axio
     Follows by set extensionality.
 \end{proof}
 
+\begin{proposition}\label{inter_subseteq}
+    Suppose $A,B\subseteq C$.
+    Then $A\inter B\subseteq C$.
+\end{proposition}
+
 \begin{abbreviation}\label{closedunderinter}
     $T$ is closed under binary intersections
     iff for every $U,V\in T$ we have $U\inter V\in T$.
diff --git a/library/set/powerset.tex b/library/set/powerset.tex
index 7f30f68..ec5866f 100644
--- a/library/set/powerset.tex
+++ b/library/set/powerset.tex
@@ -46,6 +46,17 @@
     Follows by \cref{pow_iff,unions_subseteq_of_powerset_is_subseteq}.
 \end{proof}
 
+
+\begin{proposition}\label{inter_powerset}
+    Let $A,B\in\pow{C}$.
+    Then $A\inter B\in\pow{C}$.
+\end{proposition}
+\begin{proof}
+    We have $A,B\subseteq C$ by \cref{pow_iff}.
+    $A\inter B\subseteq C$ by \cref{inter_subseteq}.
+    Follows by \cref{pow_iff}.
+\end{proof}
+
 \begin{proposition}\label{unions_powerset}
     $\unions{\pow{A}} = A$.
 \end{proposition}