Theorem inf0 9540

Description: Existence of ω implies our axiom of infinity ax-inf 9557. The proof shows that the especially contrived class "ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) " exists, is a subset of its union, and contains a given set 𝑥 (and thus is nonempty). Thus, it provides an example demonstrating that a set 𝑦 exists with the necessary properties demanded by ax-inf 9557. (Contributed by NM, 15-Oct-1996.) Revised to closed form. (Revised by BJ, 20-May-2024.)

Assertion

Ref	Expression
inf0	⊢ (ω ∈ 𝑉 → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem inf0

Dummy variables 𝑣 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fr0g 8372	. . . 4 ⊢ (𝑥 ∈ V → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) = 𝑥)
2	1	elv 3437	. . 3 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) = 𝑥
3		frfnom 8371	. . . 4 ⊢ (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω
4		peano1 7836	. . . 4 ⊢ ∅ ∈ ω
5		fnfvelrn 7028	. . . 4 ⊢ (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
6	3, 4, 5	mp2an 698	. . 3 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
7	2, 6	eqeltrri 2837	. 2 ⊢ 𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
8		fvelrnb 6894	. . . . 5 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω → (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ↔ ∃𝑓 ∈ ω ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧))
9	3, 8	ax-mp 5	. . . 4 ⊢ (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ↔ ∃𝑓 ∈ ω ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧)
10		fvex 6847	. . . . . . . . . 10 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V
11	10	sucid 6401	. . . . . . . . 9 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓)
12	10	sucex 7756	. . . . . . . . . 10 ⊢ suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V
13		eqid 2740	. . . . . . . . . . 11 ⊢ (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) = (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
14		suceq 6385	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → suc 𝑧 = suc 𝑣)
15		suceq 6385	. . . . . . . . . . 11 ⊢ (𝑧 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) → suc 𝑧 = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓))
16	13, 14, 15	frsucmpt2 8376	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ω ∧ suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V) → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓))
17	12, 16	mpan2 697	. . . . . . . . 9 ⊢ (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓))
18	11, 17	eleqtrrid 2847	. . . . . . . 8 ⊢ (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓))
19		eleq1 2828	. . . . . . . 8 ⊢ (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ↔ 𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓)))
20	18, 19	imbitrid 245	. . . . . . 7 ⊢ (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (𝑓 ∈ ω → 𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓)))
21		peano2b 7830	. . . . . . . 8 ⊢ (𝑓 ∈ ω ↔ suc 𝑓 ∈ ω)
22		fnfvelrn 7028	. . . . . . . . 9 ⊢ (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω ∧ suc 𝑓 ∈ ω) → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
23	3, 22	mpan 696	. . . . . . . 8 ⊢ (suc 𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
24	21, 23	sylbi 218	. . . . . . 7 ⊢ (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
25	20, 24	jca2 518	. . . . . 6 ⊢ (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (𝑓 ∈ ω → (𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∧ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
26		fvex 6847	. . . . . . 7 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ V
27		eleq2 2829	. . . . . . . 8 ⊢ (𝑤 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓)))
28		eleq1 2828	. . . . . . . 8 ⊢ (𝑤 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) → (𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ↔ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
29	27, 28	anbi12d 638	. . . . . . 7 ⊢ (𝑤 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)) ↔ (𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∧ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
30	26, 29	spcev 3551	. . . . . 6 ⊢ ((𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∧ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
31	25, 30	syl6com 37	. . . . 5 ⊢ (𝑓 ∈ ω → (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
32	31	rexlimiv 3134	. . . 4 ⊢ (∃𝑓 ∈ ω ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
33	9, 32	sylbi 218	. . 3 ⊢ (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
34	33	ax-gen 1802	. 2 ⊢ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
35		fndm 6595	. . . . . 6 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω → dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) = ω)
36	3, 35	ax-mp 5	. . . . 5 ⊢ dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) = ω
37		id 22	. . . . 5 ⊢ (ω ∈ 𝑉 → ω ∈ 𝑉)
38	36, 37	eqeltrid 2844	. . . 4 ⊢ (ω ∈ 𝑉 → dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ 𝑉)
39		fnfun 6592	. . . . 5 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω → Fun (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
40	3, 39	ax-mp 5	. . . 4 ⊢ Fun (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
41		funrnex 7903	. . . 4 ⊢ (dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ 𝑉 → (Fun (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V))
42	38, 40, 41	mpisyl 21	. . 3 ⊢ (ω ∈ 𝑉 → ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V)
43		eleq2 2829	. . . . 5 ⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
44		eleq2 2829	. . . . . . 7 ⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
45		eleq2 2829	. . . . . . . . 9 ⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
46	45	anbi2d 636	. . . . . . . 8 ⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
47	46	exbidv 1928	. . . . . . 7 ⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
48	44, 47	imbi12d 345	. . . . . 6 ⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ((𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)) ↔ (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))))
49	48	albidv 1927	. . . . 5 ⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)) ↔ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))))
50	43, 49	anbi12d 638	. . . 4 ⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ((𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) ↔ (𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∧ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))))
51	50	spcegv 3542	. . 3 ⊢ (ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V → ((𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∧ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))) → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))))
52	42, 51	syl 17	. 2 ⊢ (ω ∈ 𝑉 → ((𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∧ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))) → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))))
53	7, 34, 52	mp2ani 704	1 ⊢ (ω ∈ 𝑉 → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃wrex 3064  Vcvv 3432  ∅c0 4268   ↦ cmpt 5160  dom cdm 5625  ran crn 5626   ↾ cres 5627  suc csuc 6319  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  ωcom 7813  reccrdg 8345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346

This theorem is referenced by:  axinf  9563