Metamath Proof Explorer


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 )