Theorem modelac8prim 44975

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 44974.

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 44951	. . . . 5 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → 𝑧 ≠ ∅)))
2		n0abso 44959	. . . . . . . 8 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → (𝑧 ≠ ∅ ↔ ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧))
3	2	adantlr 715	. . . . . . 7 ⊢ (((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) ∧ 𝑧 ∈ 𝑀) → (𝑧 ≠ ∅ ↔ ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧))
4	3	imbi2d 340	. . . . . 6 ⊢ (((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) ∧ 𝑧 ∈ 𝑀) → ((𝑧 ∈ 𝑥 → 𝑧 ≠ ∅) ↔ (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧)))
5	4	ralbidva 3155	. . . . 5 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → 𝑧 ≠ ∅) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧)))
6	1, 5	bitrd 279	. . . 4 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧)))
7		simpl 482	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → Tr 𝑀)
8		ralabso 44951	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))))
9	7, 8	ralabsobidv 44955	. . . . . 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 675	. . . . 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 3159	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑀 (𝑧 ∈ 𝑥 → (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) ↔ (𝑧 ∈ 𝑥 → ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))))
12		impexp 450	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) ↔ (𝑧 ∈ 𝑥 → (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))))
13		df-ne 2927	. . . . . . . . . . . . 13 ⊢ (𝑧 ≠ 𝑤 ↔ ¬ 𝑧 = 𝑤)
14	13	imbi1i 349	. . . . . . . . . . . 12 ⊢ ((𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ (¬ 𝑧 = 𝑤 → (𝑧 ∩ 𝑤) = ∅))
15		disjabso 44958	. . . . . . . . . . . . 13 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → ((𝑧 ∩ 𝑤) = ∅ ↔ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))
16	15	imbi2d 340	. . . . . . . . . . . 12 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → ((¬ 𝑧 = 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤))))
17	14, 16	bitrid 283	. . . . . . . . . . 11 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → ((𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤))))
18	17	imbi2d 340	. . . . . . . . . 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 285	. . . . . . . . 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 3157	. . . . . . . 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 285	. . . . . . 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 3155	. . . . . 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 480	. . . . 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 279	. . . 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 632	. . 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 482	. . . . . 6 ⊢ ((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) → Tr 𝑀)
27		elin 3932	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦))
28	27	eubii 2579	. . . . . . . 8 ⊢ (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣(𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦))
29		trel 5225	. . . . . . . . . . . 12 ⊢ (Tr 𝑀 → ((𝑣 ∈ 𝑦 ∧ 𝑦 ∈ 𝑀) → 𝑣 ∈ 𝑀))
30	29	imp 406	. . . . . . . . . . 11 ⊢ ((Tr 𝑀 ∧ (𝑣 ∈ 𝑦 ∧ 𝑦 ∈ 𝑀)) → 𝑣 ∈ 𝑀)
31	30	anass1rs 655	. . . . . . . . . 10 ⊢ (((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) ∧ 𝑣 ∈ 𝑦) → 𝑣 ∈ 𝑀)
32	31	adantrl 716	. . . . . . . . 9 ⊢ (((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) ∧ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)) → 𝑣 ∈ 𝑀)
33	32	reueubd 3375	. . . . . . . 8 ⊢ ((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) → (∃!𝑣 ∈ 𝑀 (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ ∃!𝑣(𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)))
34	28, 33	bitr4id 290	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 ∈ 𝑀 (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)))
35		reu6 3699	. . . . . . 7 ⊢ (∃!𝑣 ∈ 𝑀 (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤))
36	34, 35	bitrdi 287	. . . . . 6 ⊢ ((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))
37	26, 36	ralabsobidv 44955	. . . . 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 652	. . . 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 3156	. . 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 344	. 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 3155	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)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃!weu 2562   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  ∃!wreu 3354   ∩ cin 3915  ∅c0 4298  Tr wtr 5216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-v 3452  df-dif 3919  df-in 3923  df-ss 3933  df-nul 4299  df-uni 4874  df-tr 5217

This theorem is referenced by:  wfac8prim  44985