Theorem dfac1

Description: Equivalence of two versions of the Axiom of Choice ax-ac . The proof uses the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by Mario Carneiro, 17-May-2015)

	Ref	Expression
Assertion	dfac1	⊢ ( CHOICE ↔ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑥 ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfac7	⊢ ( CHOICE ↔ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑧 ∃! 𝑣 ∈ 𝑧 ∃ 𝑢 ∈ 𝑦 ( 𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢 ) )
2		aceq1	⊢ ( ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑧 ∃! 𝑣 ∈ 𝑧 ∃ 𝑢 ∈ 𝑦 ( 𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢 ) ↔ ∃ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑥 ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
3	2	albii	⊢ ( ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑧 ∃! 𝑣 ∈ 𝑧 ∃ 𝑢 ∈ 𝑦 ( 𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢 ) ↔ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑥 ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
4	1 3	bitri	⊢ ( CHOICE ↔ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑥 ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )