Theorem modelac8prim 45734

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 45733.

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 45710	. . . . 5 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → 𝑧 ≠ ∅)))
2		n0abso 45718	. . . . . . . 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 3188	. . . . 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 45710	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))))
9	7, 8	ralabsobidv 45714	. . . . . 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 3192	. . . . . . . 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 2961	. . . . . . . . . . . . 13 ⊢ (𝑧 ≠ 𝑤 ↔ ¬ 𝑧 = 𝑤)
14	13	imbi1i 352	. . . . . . . . . . . 12 ⊢ ((𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ (¬ 𝑧 = 𝑤 → (𝑧 ∩ 𝑤) = ∅))
15		disjabso 45717	. . . . . . . . . . . . 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 3190	. . . . . . . 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 3188	. . . . . 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 3922	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦))
28	27	eubii 2615	. . . . . . . 8 ⊢ (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣(𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦))
29		trel 5228	. . . . . . . . . . . 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 3388	. . . . . . . 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 3691	. . . . . . 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 45714	. . . . 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 3189	. . 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 3188	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 2146  ∃!weu 2598   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  ∃!wreu 3369   ∩ cin 3905  ∅c0 4286  Tr wtr 5220

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-v 3459  df-dif 3909  df-in 3913  df-ss 3923  df-nul 4287  df-uni 4875  df-tr 5221

This theorem is used by:  wfac8prim  45744