Metamath Proof Explorer


Theorem ac6c5

Description: Equivalent of Axiom of Choice. B is a collection B ( x ) of nonempty sets. Remark after Theorem 10.46 of TakeutiZaring p. 98. (Contributed by Mario Carneiro, 22-Mar-2013)

Ref Expression
Hypotheses ac6c4.1
|- A e. _V
ac6c4.2
|- B e. _V
Assertion ac6c5
|- ( A. x e. A B =/= (/) -> E. f A. x e. A ( f ` x ) e. B )

Proof

Step Hyp Ref Expression
1 ac6c4.1
 |-  A e. _V
2 ac6c4.2
 |-  B e. _V
3 1 2 ac6c4
 |-  ( A. x e. A B =/= (/) -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. B ) )
4 exsimpr
 |-  ( E. f ( f Fn A /\ A. x e. A ( f ` x ) e. B ) -> E. f A. x e. A ( f ` x ) e. B )
5 3 4 syl
 |-  ( A. x e. A B =/= (/) -> E. f A. x e. A ( f ` x ) e. B )