# Theory Zorn_Lemma

theory Zorn_Lemma
imports Main
```(*  Title:      HOL/Hahn_Banach/Zorn_Lemma.thy
Author:     Gertrud Bauer, TU Munich
*)

section ‹Zorn's Lemma›

theory Zorn_Lemma
imports Main
begin

text ‹
Zorn's Lemmas states: if every linear ordered subset of an ordered set ‹S›
has an upper bound in ‹S›, then there exists a maximal element in ‹S›. In
our application, ‹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 ‹S› is non-empty, it
suffices to show that for every non-empty chain ‹c› in ‹S› the union of ‹c›
also lies in ‹S›.
›

theorem Zorn's_Lemma:
assumes r: "⋀c. c ∈ chains S ⟹ ∃x. x ∈ c ⟹ ⋃c ∈ S"
and aS: "a ∈ S"
shows "∃y ∈ S. ∀z ∈ S. y ⊆ z ⟶ z = y"
proof (rule Zorn_Lemma2)
show "∀c ∈ chains S. ∃y ∈ S. ∀z ∈ c. z ⊆ y"
proof
fix c assume "c ∈ chains S"
show "∃y ∈ S. ∀z ∈ c. z ⊆ y"
proof cases
txt ‹If ‹c› is an empty chain, then every element in ‹S› is an upper
bound of ‹c›.›

assume "c = {}"
with aS show ?thesis by fast

txt ‹If ‹c› is non-empty, then ‹⋃c› is an upper bound of ‹c›, lying in
‹S›.›
next
assume "c ≠ {}"
show ?thesis
proof
show "∀z ∈ c. z ⊆ ⋃c" by fast
show "⋃c ∈ S"
proof (rule r)
from ‹c ≠ {}› show "∃x. x ∈ c" by fast
show "c ∈ chains S" by fact
qed
qed
qed
qed
qed

end
```