Description: Generalization of the Axiom of Choice to proper classes. B is a collection B ( x ) of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006)
Ref | Expression | ||
---|---|---|---|
Hypothesis | ac6s4.1 | |- A e. _V |
|
Assertion | ac6s4 | |- ( A. x e. A B =/= (/) -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ac6s4.1 | |- A e. _V |
|
2 | n0 | |- ( B =/= (/) <-> E. y y e. B ) |
|
3 | 2 | ralbii | |- ( A. x e. A B =/= (/) <-> A. x e. A E. y y e. B ) |
4 | eleq1 | |- ( y = ( f ` x ) -> ( y e. B <-> ( f ` x ) e. B ) ) |
|
5 | 1 4 | ac6s2 | |- ( A. x e. A E. y y e. B -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. B ) ) |
6 | 3 5 | sylbi | |- ( A. x e. A B =/= (/) -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. B ) ) |