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theory ZornLemma(* Title: HOL/Real/HahnBanach/ZornLemma.thy ID: $Id: ZornLemma.thy,v 1.12 2005/06/17 14:13:10 haftmann Exp $ Author: Gertrud Bauer, TU Munich *) header {* Zorn's Lemma *} theory ZornLemma imports Zorn begin text {* Zorn's Lemmas states: if every linear ordered subset of an ordered set @{text S} has an upper bound in @{text S}, then there exists a maximal element in @{text S}. In our application, @{text S} is a set of sets ordered by set inclusion. Since the union of a chain of sets is an upper bound for all elements of the chain, the conditions of Zorn's lemma can be modified: if @{text S} is non-empty, it suffices to show that for every non-empty chain @{text c} in @{text S} the union of @{text c} also lies in @{text S}. *} theorem Zorn's_Lemma: assumes r: "!!c. c ∈ chain S ==> ∃x. x ∈ c ==> \<Union>c ∈ S" and aS: "a ∈ S" shows "∃y ∈ S. ∀z ∈ S. y ⊆ z --> y = z" proof (rule Zorn_Lemma2) txt_raw {* \footnote{See \url{http://isabelle.in.tum.de/library/HOL/HOL-Complex/Zorn.html}} \isanewline *} show "∀c ∈ chain S. ∃y ∈ S. ∀z ∈ c. z ⊆ y" proof fix c assume "c ∈ chain S" show "∃y ∈ S. ∀z ∈ c. z ⊆ y" proof cases txt {* If @{text c} is an empty chain, then every element in @{text S} is an upper bound of @{text c}. *} assume "c = {}" with aS show ?thesis by fast txt {* If @{text c} is non-empty, then @{text "\<Union>c"} is an upper bound of @{text c}, lying in @{text S}. *} next assume c: "c ≠ {}" show ?thesis proof show "∀z ∈ c. z ⊆ \<Union>c" by fast show "\<Union>c ∈ S" proof (rule r) from c show "∃x. x ∈ c" by fast qed qed qed qed qed end
theorem Zorn's_Lemma:
[| !!c. [| c ∈ chain S; ∃x. x ∈ c |] ==> Union c ∈ S; a ∈ S |] ==> ∃y∈S. ∀z∈S. y ⊆ z --> y = z