Theorem modelac8prim 46246

Description: If 𝑀 is a transitive class, then the following are equivalent. (1) Every nonempty set 𝑥 ∈ 𝑀 of pairwise disjoint nonempty sets has a choice set in 𝑀. (2) The class 𝑀 models the Axiom of Choice, in the form ac8prim 46245.

Lemma II.2.11(7) of [Kunen2] p. 114. Kunen has the additional hypotheses that the Extensionality, Separation, Pairing, and Union axioms are true in 𝑀. This, apparently, is because Kunen's statement of the Axiom of Choice uses defined notions, including ∅ and ∩, and these axioms guarantee that these notions are well-defined. When we state the axiom using primitives only, the need for these hypotheses disappears. (Contributed by Eric Schmidt, 19-Oct-2025.)

Assertion

Ref	Expression
modelac8prim	⊢ (Tr 𝑀 → (∀𝑥 ∈ 𝑀 ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝑀 ((∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))))

Distinct variable group:   𝑥,𝑧,𝑦,𝑤,𝑣,𝑀

Proof of Theorem modelac8prim

Step	Hyp	Ref	Expression
1		ralabso 46222	. . . . 5 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → 𝑧 ≠ ∅)))
2		n0abso 46230	. . . . . . . 8 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → (𝑧 ≠ ∅ ↔ ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧))
3	2	adantlr 728	. . . . . . 7 ⊢ (((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) ∧ 𝑧 ∈ 𝑀) → (𝑧 ≠ ∅ ↔ ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧))
4	3	imbi2d 343	. . . . . 6 ⊢ (((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) ∧ 𝑧 ∈ 𝑀) → ((𝑧 ∈ 𝑥 → 𝑧 ≠ ∅) ↔ (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧)))
5	4	ralbidva 3187	. . . . 5 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → 𝑧 ≠ ∅) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧)))
6	1, 5	bitrd 282	. . . 4 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧)))
7		simpl 488	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → Tr 𝑀)
8		ralabso 46222	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))))
9	7, 8	ralabsobidv 46226	. . . . . 6 ⊢ (((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))))
10	9	anabss3 688	. . . . 5 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))))
11		r19.21v 3191	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑀 (𝑧 ∈ 𝑥 → (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) ↔ (𝑧 ∈ 𝑥 → ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))))
12		impexp 456	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) ↔ (𝑧 ∈ 𝑥 → (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))))
13		df-ne 2960	. . . . . . . . . . . . 13 ⊢ (𝑧 ≠ 𝑤 ↔ ¬ 𝑧 = 𝑤)
14	13	imbi1i 352	. . . . . . . . . . . 12 ⊢ ((𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ (¬ 𝑧 = 𝑤 → (𝑧 ∩ 𝑤) = ∅))
15		disjabso 46229	. . . . . . . . . . . . 13 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → ((𝑧 ∩ 𝑤) = ∅ ↔ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))
16	15	imbi2d 343	. . . . . . . . . . . 12 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → ((¬ 𝑧 = 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤))))
17	14, 16	bitrid 286	. . . . . . . . . . 11 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → ((𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤))))
18	17	imbi2d 343	. . . . . . . . . 10 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → (((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) ↔ ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))))
19	12, 18	bitr3id 288	. . . . . . . . 9 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → ((𝑧 ∈ 𝑥 → (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) ↔ ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))))
20	19	ralbidv 3189	. . . . . . . 8 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → (∀𝑤 ∈ 𝑀 (𝑧 ∈ 𝑥 → (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) ↔ ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))))
21	11, 20	bitr3id 288	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → ((𝑧 ∈ 𝑥 → ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) ↔ ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))))
22	21	ralbidva 3187	. . . . . 6 ⊢ (Tr 𝑀 → (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) ↔ ∀𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))))
23	22	adantr 486	. . . . 5 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) ↔ ∀𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))))
24	10, 23	bitrd 282	. . . 4 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ ∀𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))))
25	6, 24	anbi12d 644	. . 3 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) ↔ (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤))))))
26		simpl 488	. . . . . 6 ⊢ ((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) → Tr 𝑀)
27		elin 4096	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦))
28	27	eubii 2613	. . . . . . . 8 ⊢ (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣(𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦))
29		trel 5226	. . . . . . . . . . . 12 ⊢ (Tr 𝑀 → ((𝑣 ∈ 𝑦 ∧ 𝑦 ∈ 𝑀) → 𝑣 ∈ 𝑀))
30	29	imp 412	. . . . . . . . . . 11 ⊢ ((Tr 𝑀 ∧ (𝑣 ∈ 𝑦 ∧ 𝑦 ∈ 𝑀)) → 𝑣 ∈ 𝑀)
31	30	anass1rs 668	. . . . . . . . . 10 ⊢ (((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) ∧ 𝑣 ∈ 𝑦) → 𝑣 ∈ 𝑀)
32	31	adantrl 729	. . . . . . . . 9 ⊢ (((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) ∧ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)) → 𝑣 ∈ 𝑀)
33	32	reueubd 3386	. . . . . . . 8 ⊢ ((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) → (∃!𝑣 ∈ 𝑀 (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ ∃!𝑣(𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)))
34	28, 33	bitr4id 293	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 ∈ 𝑀 (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)))
35		reu6 3687	. . . . . . 7 ⊢ (∃!𝑣 ∈ 𝑀 (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤))
36	34, 35	bitrdi 290	. . . . . 6 ⊢ ((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))
37	26, 36	ralabsobidv 46226	. . . . 5 ⊢ (((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤))))
38	37	an32s 665	. . . 4 ⊢ (((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) ∧ 𝑦 ∈ 𝑀) → (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤))))
39	38	rexbidva 3188	. . 3 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤))))
40	25, 39	imbi12d 347	. 2 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ((∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))))
41	40	ralbidva 3187	1 ⊢ (Tr 𝑀 → (∀𝑥 ∈ 𝑀 ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝑀 ((∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃!weu 2596   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  ∃!wreu 3367   ∩ cin 4092  ∅c0 4281  Tr wtr 5218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-nul 4282  df-uni 4873  df-tr 5219

This theorem is used by:  wfac8prim  46256