Theorem modelac8prim 45588

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 45587.

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 45564	. . . . 5 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → 𝑧 ≠ ∅)))
2		n0abso 45572	. . . . . . . 8 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → (𝑧 ≠ ∅ ↔ ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧))
3	2	adantlr 727	. . . . . . 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 3192	. . . . 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 487	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → Tr 𝑀)
8		ralabso 45564	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))))
9	7, 8	ralabsobidv 45568	. . . . . 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 687	. . . . 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 3196	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑀 (𝑧 ∈ 𝑥 → (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) ↔ (𝑧 ∈ 𝑥 → ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))))
12		impexp 455	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) ↔ (𝑧 ∈ 𝑥 → (𝑤 ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))))
13		df-ne 2965	. . . . . . . . . . . . 13 ⊢ (𝑧 ≠ 𝑤 ↔ ¬ 𝑧 = 𝑤)
14	13	imbi1i 352	. . . . . . . . . . . 12 ⊢ ((𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ (¬ 𝑧 = 𝑤 → (𝑧 ∩ 𝑤) = ∅))
15		disjabso 45571	. . . . . . . . . . . . 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 3194	. . . . . . . 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 3192	. . . . . 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 485	. . . . 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 643	. . 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 487	. . . . . 6 ⊢ ((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) → Tr 𝑀)
27		elin 3929	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦))
28	27	eubii 2619	. . . . . . . 8 ⊢ (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣(𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦))
29		trel 5227	. . . . . . . . . . . 12 ⊢ (Tr 𝑀 → ((𝑣 ∈ 𝑦 ∧ 𝑦 ∈ 𝑀) → 𝑣 ∈ 𝑀))
30	29	imp 411	. . . . . . . . . . 11 ⊢ ((Tr 𝑀 ∧ (𝑣 ∈ 𝑦 ∧ 𝑦 ∈ 𝑀)) → 𝑣 ∈ 𝑀)
31	30	anass1rs 667	. . . . . . . . . 10 ⊢ (((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) ∧ 𝑣 ∈ 𝑦) → 𝑣 ∈ 𝑀)
32	31	adantrl 728	. . . . . . . . 9 ⊢ (((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) ∧ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦)) → 𝑣 ∈ 𝑀)
33	32	reueubd 3393	. . . . . . . 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 3698	. . . . . . 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 45568	. . . . 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 664	. . . 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 3193	. . 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 3192	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 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃!weu 2602   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  ∃!wreu 3374   ∩ cin 3912  ∅c0 4294  Tr wtr 5219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-v 3465  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4295  df-uni 4874  df-tr 5220

This theorem is referenced by:  wfac8prim  45598