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 ) |
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 ) |