equal
deleted
inserted
replaced
1809 lemma card_Suc_eq: |
1809 lemma card_Suc_eq: |
1810 "card A = Suc k \<longleftrightarrow> |
1810 "card A = Suc k \<longleftrightarrow> |
1811 (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))" |
1811 (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))" |
1812 by (auto simp: card_insert_if card_gt_0_iff elim!: card_eq_SucD) |
1812 by (auto simp: card_insert_if card_gt_0_iff elim!: card_eq_SucD) |
1813 |
1813 |
|
1814 lemma card_Suc_eq_finite: |
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1815 "card A = Suc k \<longleftrightarrow> (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> finite B)" |
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1816 unfolding card_Suc_eq using card_gt_0_iff by fastforce |
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1817 |
1814 lemma card_1_singletonE: |
1818 lemma card_1_singletonE: |
1815 assumes "card A = 1" |
1819 assumes "card A = 1" |
1816 obtains x where "A = {x}" |
1820 obtains x where "A = {x}" |
1817 using assms by (auto simp: card_Suc_eq) |
1821 using assms by (auto simp: card_Suc_eq) |
1818 |
1822 |