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 ⊢ ∀ z ∈ x z ≠ ∅ ∧ ∀ z ∈ x ∀ w ∈ x z ≠ w → z ∩ w = ∅ → ∃ y ∀ z ∈ x ∃! v v ∈ z ∩ y

Proof

Step Hyp Ref Expression
1 dfac5 ⊢ CHOICE ↔ ∀ x ∀ z ∈ x z ≠ ∅ ∧ ∀ z ∈ x ∀ w ∈ x z ≠ w → z ∩ w = ∅ → ∃ y ∀ z ∈ x ∃! v v ∈ z ∩ y
2 1 axaci ⊢ ∀ z ∈ x z ≠ ∅ ∧ ∀ z ∈ x ∀ w ∈ x z ≠ w → z ∩ w = ∅ → ∃ y ∀ z ∈ x ∃! v v ∈ z ∩ y