Description: Variant of Zorn's lemma zorn in which (/) , the union of the empty chain, is not required to be an element of A . (Contributed by Jeff Madsen, 5-Jan-2011)
Ref | Expression | ||
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Hypothesis | zornn0.1 | |- A e. _V |
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Assertion | zornn0 | |- ( ( A =/= (/) /\ A. z ( ( z C_ A /\ z =/= (/) /\ [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 ) |
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3 | 1 2 | ax-mp | |- A e. dom card |
4 | zornn0g | |- ( ( A e. dom card /\ A =/= (/) /\ A. z ( ( z C_ A /\ z =/= (/) /\ [C.] Or z ) -> U. z e. A ) ) -> E. x e. A A. y e. A -. x C. y ) |
|
5 | 3 4 | mp3an1 | |- ( ( A =/= (/) /\ A. z ( ( z C_ A /\ z =/= (/) /\ [C.] Or z ) -> U. z e. A ) ) -> E. x e. A A. y e. A -. x C. y ) |