diff options
| -rw-r--r-- | library/set.tex | 3 | ||||
| -rw-r--r-- | library/topology/basis.tex | 2 |
2 files changed, 4 insertions, 1 deletions
diff --git a/library/set.tex b/library/set.tex index 2fd18ea..69b9526 100644 --- a/library/set.tex +++ b/library/set.tex @@ -654,6 +654,9 @@ The $\operatorname{\textsf{cons}}$ operation is determined by the following axio Suppose $(A\inter B)\union C = A\inter (B\union C)$. Then $C\subseteq A$. \end{proposition} +\begin{proof} + Follows by \cref{union_upper_right,union_upper_left,subseteq_union_iff,subseteq_antisymmetric,subseteq_inter_iff}. +\end{proof} % From Isabelle/ZF equalities theory \begin{proposition}\label{union_inter_crazy} diff --git a/library/topology/basis.tex b/library/topology/basis.tex index bd7d15a..052c551 100644 --- a/library/topology/basis.tex +++ b/library/topology/basis.tex @@ -64,7 +64,7 @@ Then $\unions{F}\in\genOpens{B}{X}$. \end{lemma} \begin{proof} - We have $\unions{F} \in \pow{X}$. + We have $\unions{F} \in \pow{X}$ by \cref{genopens,subseteq,pow_iff,unions_family,powerset_elim}. Show for all $x\in \unions{F}$ there exists $W \in B$ such that $x\in W$ and $W \subseteq \unions{F}$. |