Theorem zorn

Description: Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of Enderton p. 151. See zorn2 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004)

	Ref	Expression
Hypothesis	zornn0.1	|- A e. _V
Assertion	zorn	|- ( A. z ( ( z C_ A /\ [C.] Or z ) -> U. z e. A ) -> E. x e. A A. y e. A -. x C. y )

Proof

Step	Hyp	Ref	Expression
1		zornn0.1	|- A e. _V
2		numth3	|- ( A e. _V -> A e. dom card )
3	1 2	ax-mp	|- A e. dom card
4		zorng	|- ( ( A e. dom card /\ A. z ( ( z C_ A /\ [C.] Or z ) -> U. z e. A ) ) -> E. x e. A A. y e. A -. x C. y )
5	3 4	mpan	|- ( A. z ( ( z C_ A /\ [C.] Or z ) -> U. z e. A ) -> E. x e. A A. y e. A -. x C. y )