Metamath Proof Explorer

Theorem ac8

Description: An Axiom of Choice equivalent. Given a family x of mutually disjoint nonempty sets, there exists a set y containing exactly one member from each set in the family. Theorem 6M(4) of Enderton p. 151. (Contributed by NM, 14-May-2004)

		Ref	Expression
	Assertion	ac8	\|- ( ( A. z e. x z =/= (/) /\ A. z e. x A. w e. x ( z =/= w -> ( z i^i w ) = (/) ) ) -> E. y A. z e. x E! v v e. ( z i^i y ) )

Proof

Step	Hyp	Ref	Expression
1		dfac5	\|- ( CHOICE <-> A. x ( ( A. z e. x z =/= (/) /\ A. z e. x A. w e. x ( z =/= w -> ( z i^i w ) = (/) ) ) -> E. y A. z e. x E! v v e. ( z i^i y ) ) )
2	1	axaci	\|- ( ( A. z e. x z =/= (/) /\ A. z e. x A. w e. x ( z =/= w -> ( z i^i w ) = (/) ) ) -> E. y A. z e. x E! v v e. ( z i^i y ) )

Step

Hyp

Ref

Expression

dfac5

 |-  ( CHOICE <-> A. x ( ( A. z e. x z =/= (/) /\ A. z e. x A. w e. x ( z =/= w -> ( z i^i w ) = (/) ) ) -> E. y A. z e. x E! v v e. ( z i^i y ) ) )

axaci

 |-  ( ( A. z e. x z =/= (/) /\ A. z e. x A. w e. x ( z =/= w -> ( z i^i w ) = (/) ) ) -> E. y A. z e. x E! v v e. ( z i^i y ) )