Metamath Proof Explorer


Theorem ac6s4

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 ) )

Proof

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 ) )