Theorem dfackm

Description: Equivalence of the Axiom of Choice and Maes' AC ackm . The proof consists of lemmas kmlem1 through kmlem16 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e., replacing dfac5 with biid ) establishes the AC equivalence shown by Maes' writeup. The left-hand-side AC shown here was chosen because it is shorter to display. (Contributed by NM, 13-Apr-2004) (Revised by Mario Carneiro, 17-May-2015)

	Ref	Expression
Assertion	dfackm	⊢ ( CHOICE ↔ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∃ 𝑣 ∀ 𝑢 ( ( 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑦 → ( ( 𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣 ) ∧ 𝑧 ∈ 𝑣 ) ) ) ∨ ( ¬ 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑥 → ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ∧ ( ( 𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦 ) → 𝑢 = 𝑣 ) ) ) ) ) )

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		eqid	⊢ { 𝑡 ∣ ∃ ℎ ∈ 𝑥 𝑡 = ( ℎ ∖ ∪ ( 𝑥 ∖ { ℎ } ) ) } = { 𝑡 ∣ ∃ ℎ ∈ 𝑥 𝑡 = ( ℎ ∖ ∪ ( 𝑥 ∖ { ℎ } ) ) }
3	2	kmlem13	⊢ ( ∀ 𝑥 ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ↔ ∀ 𝑥 ( ¬ ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ) )
4		kmlem8	⊢ ( ( ¬ ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ) ↔ ( ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ∨ ∃ 𝑦 ( ¬ 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ) )
5	4	albii	⊢ ( ∀ 𝑥 ( ¬ ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ) ↔ ∀ 𝑥 ( ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ∨ ∃ 𝑦 ( ¬ 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ) )
6	3 5	bitri	⊢ ( ∀ 𝑥 ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ↔ ∀ 𝑥 ( ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ∨ ∃ 𝑦 ( ¬ 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ) )
7		df-ne	⊢ ( 𝑦 ≠ 𝑣 ↔ ¬ 𝑦 = 𝑣 )
8	7	bicomi	⊢ ( ¬ 𝑦 = 𝑣 ↔ 𝑦 ≠ 𝑣 )
9	8	anbi2i	⊢ ( ( 𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣 ) ↔ ( 𝑣 ∈ 𝑥 ∧ 𝑦 ≠ 𝑣 ) )
10	9	anbi1i	⊢ ( ( ( 𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣 ) ∧ 𝑧 ∈ 𝑣 ) ↔ ( ( 𝑣 ∈ 𝑥 ∧ 𝑦 ≠ 𝑣 ) ∧ 𝑧 ∈ 𝑣 ) )
11	10	imbi2i	⊢ ( ( 𝑧 ∈ 𝑦 → ( ( 𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣 ) ∧ 𝑧 ∈ 𝑣 ) ) ↔ ( 𝑧 ∈ 𝑦 → ( ( 𝑣 ∈ 𝑥 ∧ 𝑦 ≠ 𝑣 ) ∧ 𝑧 ∈ 𝑣 ) ) )
12		biid	⊢ ( ( 𝑧 ∈ 𝑥 → ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ∧ ( ( 𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦 ) → 𝑢 = 𝑣 ) ) ) ↔ ( 𝑧 ∈ 𝑥 → ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ∧ ( ( 𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦 ) → 𝑢 = 𝑣 ) ) ) )
13		biid	⊢ ( ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ↔ ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) )
14	11 12 13	kmlem16	⊢ ( ( ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ∨ ∃ 𝑦 ( ¬ 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ) ↔ ∃ 𝑦 ∀ 𝑧 ∃ 𝑣 ∀ 𝑢 ( ( 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑦 → ( ( 𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣 ) ∧ 𝑧 ∈ 𝑣 ) ) ) ∨ ( ¬ 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑥 → ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ∧ ( ( 𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦 ) → 𝑢 = 𝑣 ) ) ) ) ) )
15	14	albii	⊢ ( ∀ 𝑥 ( ∃ 𝑧 ∈ 𝑥 ∀ 𝑣 ∈ 𝑧 ∃ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 ∧ 𝑣 ∈ ( 𝑧 ∩ 𝑤 ) ) ∨ ∃ 𝑦 ( ¬ 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ) ↔ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∃ 𝑣 ∀ 𝑢 ( ( 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑦 → ( ( 𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣 ) ∧ 𝑧 ∈ 𝑣 ) ) ) ∨ ( ¬ 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑥 → ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ∧ ( ( 𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦 ) → 𝑢 = 𝑣 ) ) ) ) ) )
16	6 15	bitri	⊢ ( ∀ 𝑥 ( ( ∀ 𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑥 ( 𝑧 ≠ 𝑤 → ( 𝑧 ∩ 𝑤 ) = ∅ ) ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ∃! 𝑣 𝑣 ∈ ( 𝑧 ∩ 𝑦 ) ) ↔ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∃ 𝑣 ∀ 𝑢 ( ( 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑦 → ( ( 𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣 ) ∧ 𝑧 ∈ 𝑣 ) ) ) ∨ ( ¬ 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑥 → ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ∧ ( ( 𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦 ) → 𝑢 = 𝑣 ) ) ) ) ) )
17	1 16	bitri	⊢ ( CHOICE ↔ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∃ 𝑣 ∀ 𝑢 ( ( 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑦 → ( ( 𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣 ) ∧ 𝑧 ∈ 𝑣 ) ) ) ∨ ( ¬ 𝑦 ∈ 𝑥 ∧ ( 𝑧 ∈ 𝑥 → ( ( 𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦 ) ∧ ( ( 𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦 ) → 𝑢 = 𝑣 ) ) ) ) ) )