--- a/src/ZF/Cardinal_AC.thy Thu Mar 15 23:06:22 2012 +0100
+++ b/src/ZF/Cardinal_AC.thy Fri Mar 16 16:29:28 2012 +0000
@@ -134,29 +134,38 @@
(*Kunen's Lemma 10.21*)
lemma cardinal_UN_le:
- "[| InfCard(K); \<forall>i\<in>K. |X(i)| \<le> K |] ==> |\<Union>i\<in>K. X(i)| \<le> K"
-apply (simp add: InfCard_is_Card le_Card_iff)
-apply (rule lepoll_trans)
- prefer 2
- apply (rule InfCard_square_eq [THEN eqpoll_imp_lepoll])
- apply (simp add: InfCard_is_Card Card_cardinal_eq)
-apply (unfold lepoll_def)
-apply (frule InfCard_is_Card [THEN Card_is_Ord])
-apply (erule AC_ball_Pi [THEN exE])
-apply (rule exI)
-(*Lemma needed in both subgoals, for a fixed z*)
-apply (subgoal_tac "\<forall>z\<in>(\<Union>i\<in>K. X (i)). z: X (LEAST i. z:X (i)) &
- (LEAST i. z:X (i)) \<in> K")
- prefer 2
- apply (fast intro!: Least_le [THEN lt_trans1, THEN ltD] ltI
- elim!: LeastI Ord_in_Ord)
-apply (rule_tac c = "%z. <LEAST i. z:X (i), f ` (LEAST i. z:X (i)) ` z>"
- and d = "%<i,j>. converse (f`i) ` j" in lam_injective)
-(*Instantiate the lemma proved above*)
-by (blast intro: inj_is_fun [THEN apply_type] dest: apply_type, force)
+ assumes K: "InfCard(K)"
+ shows "(!!i. i\<in>K ==> |X(i)| \<le> K) ==> |\<Union>i\<in>K. X(i)| \<le> K"
+proof (simp add: K InfCard_is_Card le_Card_iff)
+ have [intro]: "Ord(K)" by (blast intro: InfCard_is_Card Card_is_Ord K)
+ assume "!!i. i\<in>K ==> X(i) \<lesssim> K"
+ hence "!!i. i\<in>K ==> \<exists>f. f \<in> inj(X(i), K)" by (simp add: lepoll_def)
+ with AC_Pi obtain f where f: "f \<in> (\<Pi> i\<in>K. inj(X(i), K))"
+ apply - apply atomize apply auto done
+ { fix z
+ assume z: "z \<in> (\<Union>i\<in>K. X(i))"
+ then obtain i where i: "i \<in> K" "Ord(i)" "z \<in> X(i)"
+ by (blast intro: Ord_in_Ord [of K])
+ hence "(LEAST i. z \<in> X(i)) \<le> i" by (fast intro: Least_le)
+ hence "(LEAST i. z \<in> X(i)) < K" by (best intro: lt_trans1 ltI i)
+ hence "(LEAST i. z \<in> X(i)) \<in> K" and "z \<in> X(LEAST i. z \<in> X(i))"
+ by (auto intro: LeastI ltD i)
+ } note mems = this
+ have "(\<Union>i\<in>K. X(i)) \<lesssim> K \<times> K"
+ proof (unfold lepoll_def)
+ show "\<exists>f. f \<in> inj(\<Union>RepFun(K, X), K \<times> K)"
+ apply (rule exI)
+ apply (rule_tac c = "%z. \<langle>LEAST i. z \<in> X(i), f ` (LEAST i. z \<in> X(i)) ` z\<rangle>"
+ and d = "%\<langle>i,j\<rangle>. converse (f`i) ` j" in lam_injective)
+ apply (force intro: f inj_is_fun mems apply_type Perm.left_inverse)+
+ done
+ qed
+ also have "... \<approx> K"
+ by (simp add: K InfCard_square_eq InfCard_is_Card Card_cardinal_eq)
+ finally show "(\<Union>i\<in>K. X(i)) \<lesssim> K" .
+qed
-
-(*The same again, using csucc*)
+text{*The same again, using @{term csucc}*}
lemma cardinal_UN_lt_csucc:
"[| InfCard(K); \<forall>i\<in>K. |X(i)| < csucc(K) |]
==> |\<Union>i\<in>K. X(i)| < csucc(K)"