1 (* Title: HOL/Real/HahnBanach/ZornLemma.thy
3 Author: Gertrud Bauer, TU Munich
6 header {* Zorn's Lemma *}
13 Zorn's Lemmas states: if every linear ordered subset of an ordered
14 set @{text S} has an upper bound in @{text S}, then there exists a
15 maximal element in @{text S}. In our application, @{text S} is a
16 set of sets ordered by set inclusion. Since the union of a chain of
17 sets is an upper bound for all elements of the chain, the conditions
18 of Zorn's lemma can be modified: if @{text S} is non-empty, it
19 suffices to show that for every non-empty chain @{text c} in @{text
20 S} the union of @{text c} also lies in @{text S}.
24 assumes r: "\<And>c. c \<in> chain S \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> \<Union>c \<in> S"
26 shows "\<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z \<longrightarrow> y = z"
27 proof (rule Zorn_Lemma2)
28 show "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"
30 fix c assume "c \<in> chain S"
31 show "\<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"
34 txt {* If @{text c} is an empty chain, then every element in
35 @{text S} is an upper bound of @{text c}. *}
38 with aS show ?thesis by fast
40 txt {* If @{text c} is non-empty, then @{text "\<Union>c"} is an upper
41 bound of @{text c}, lying in @{text S}. *}
44 assume "c \<noteq> {}"
47 show "\<forall>z \<in> c. z \<subseteq> \<Union>c" by fast
48 show "\<Union>c \<in> S"
50 from `c \<noteq> {}` show "\<exists>x. x \<in> c" by fast
51 show "c \<in> chain S" by fact