Metamath Proof Explorer

Theorem zorn2

Description: Zorn's Lemma of Monk1 p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set A (with an ordering relation R ) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 through zorn2lem7 ; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 . (Contributed by NM, 6-Apr-1997) (Revised by Mario Carneiro, 9-May-2015)

|- A e. _V

|- ( ( R Po A /\ A. w ( ( w C_ A /\ R Or w ) -> E. x e. A A. z e. w ( z R x \/ z = x ) ) ) -> E. x e. A A. y e. A -. x R y )

 |-  A e. _V

 |-  ( A e. _V -> A e. dom card )

 |-  A e. dom card

 |-  ( ( A e. dom card /\ R Po A /\ A. w ( ( w C_ A /\ R Or w ) -> E. x e. A A. z e. w ( z R x \/ z = x ) ) ) -> E. x e. A A. y e. A -. x R y )

 |-  ( ( R Po A /\ A. w ( ( w C_ A /\ R Or w ) -> E. x e. A A. z e. w ( z R x \/ z = x ) ) ) -> E. x e. A A. y e. A -. x R y )