header {* Zorn's Lemma *}
theory Zorn_Lemma
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)
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 ≠ {}"
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
show "c ∈ chain S" by fact
qed
qed
qed
qed
qed
end