Metamath Proof Explorer


Theorem ac6s5

Description: Generalization of the Axiom of Choice to proper classes. B is a collection B ( x ) of nonempty, possible proper classes. Remark after Theorem 10.46 of TakeutiZaring p. 98. (Contributed by NM, 27-Mar-2006)

Ref Expression
Hypothesis ac6s4.1
|- A e. _V
Assertion ac6s5
|- ( A. x e. A B =/= (/) -> E. f A. x e. A ( f ` x ) e. B )

Proof

Step Hyp Ref Expression
1 ac6s4.1
 |-  A e. _V
2 1 ac6s4
 |-  ( A. x e. A B =/= (/) -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. B ) )
3 exsimpr
 |-  ( E. f ( f Fn A /\ A. x e. A ( f ` x ) e. B ) -> E. f A. x e. A ( f ` x ) e. B )
4 2 3 syl
 |-  ( A. x e. A B =/= (/) -> E. f A. x e. A ( f ` x ) e. B )