--- a/src/HOL/Finite_Set.thy Thu Dec 17 13:49:36 2009 -0800
+++ b/src/HOL/Finite_Set.thy Thu Dec 17 13:51:50 2009 -0800
@@ -2067,6 +2067,9 @@
lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
by auto
+lemma card_gt_0_iff: "(0 < card A) = (A \<noteq> {} & finite A)"
+ by (simp add: neq0_conv [symmetric] card_eq_0_iff)
+
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
apply(simp del:insert_Diff_single)
@@ -2123,6 +2126,14 @@
"finite B ==> B <= A ==> card (A - B) = card A - card B"
by(simp add:card_def setsum_diff_nat)
+lemma card_Diff_subset_Int:
+ assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
+proof -
+ have "A - B = A - A \<inter> B" by auto
+ thus ?thesis
+ by (simp add: card_Diff_subset AB)
+qed
+
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
apply (rule Suc_less_SucD)
apply (simp add: card_Suc_Diff1 del:card_Diff_insert)