--- a/src/HOL/Set.thy Tue Sep 06 11:31:01 2011 +0200
+++ b/src/HOL/Set.thy Tue Sep 06 14:25:16 2011 +0200
@@ -785,6 +785,26 @@
lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"
by auto
+lemma insert_eq_iff: assumes "a \<notin> A" "b \<notin> B"
+shows "insert a A = insert b B \<longleftrightarrow>
+ (if a=b then A=B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
+ (is "?L \<longleftrightarrow> ?R")
+proof
+ assume ?L
+ show ?R
+ proof cases
+ assume "a=b" with assms `?L` show ?R by (simp add: insert_ident)
+ next
+ assume "a\<noteq>b"
+ let ?C = "A - {b}"
+ have "A = insert b ?C \<and> b \<notin> ?C \<and> B = insert a ?C \<and> a \<notin> ?C"
+ using assms `?L` `a\<noteq>b` by auto
+ thus ?R using `a\<noteq>b` by auto
+ qed
+next
+ assume ?R thus ?L by(auto split: if_splits)
+qed
+
subsubsection {* Singletons, using insert *}
lemma singletonI [intro!,no_atp]: "a : {a}"