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