Theorem inf0 8817

Description: Our Axiom of Infinity derived from existence of omega. 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 8834. (Contributed by NM, 15-Oct-1996.)

Hypothesis

Ref	Expression
inf0.1	⊢ ω ∈ V

Assertion

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

Distinct variable group:   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 7816	. . . 4 ⊢ (𝑥 ∈ V → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) = 𝑥)
2	1	elv 3402	. . 3 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) = 𝑥
3		frfnom 7815	. . . 4 ⊢ (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω
4		peano1 7365	. . . 4 ⊢ ∅ ∈ ω
5		fnfvelrn 6622	. . . 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 682	. . 3 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
7	2, 6	eqeltrri 2856	. 2 ⊢ 𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
8		fvelrnb 6505	. . . . 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 6461	. . . . . . . . . 10 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V
11	10	sucid 6057	. . . . . . . . 9 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓)
12	10	sucex 7291	. . . . . . . . . 10 ⊢ suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V
13		eqid 2778	. . . . . . . . . . 11 ⊢ (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) = (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
14		suceq 6043	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → suc 𝑧 = suc 𝑣)
15		suceq 6043	. . . . . . . . . . 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 7820	. . . . . . . . . 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 681	. . . . . . . . 9 ⊢ (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓))
18	11, 17	syl5eleqr 2866	. . . . . . . 8 ⊢ (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓))
19		eleq1 2847	. . . . . . . 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	syl5ib 236	. . . . . . 7 ⊢ (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (𝑓 ∈ ω → 𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓)))
21		peano2b 7361	. . . . . . . 8 ⊢ (𝑓 ∈ ω ↔ suc 𝑓 ∈ ω)
22		fnfvelrn 6622	. . . . . . . . 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 680	. . . . . . . 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 209	. . . . . . 7 ⊢ (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
25	20, 24	jca2 509	. . . . . 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 6461	. . . . . . 7 ⊢ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ V
27		eleq2 2848	. . . . . . . 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 2847	. . . . . . . 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 624	. . . . . . 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 3502	. . . . . 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 3209	. . . 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 209	. . 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 1839	. 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 6237	. . . . . 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		inf0.1	. . . . 5 ⊢ ω ∈ V
38	36, 37	eqeltri 2855	. . . 4 ⊢ dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V
39		fnfun 6235	. . . . 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 7414	. . . 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	mp2 9	. . 3 ⊢ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V
43		eleq2 2848	. . . 4 ⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
44		eleq2 2848	. . . . . 6 ⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
45		eleq2 2848	. . . . . . . 8 ⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
46	45	anbi2d 622	. . . . . . 7 ⊢ (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 1964	. . . . . 6 ⊢ (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 336	. . . . 5 ⊢ (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 1963	. . . 4 ⊢ (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 624	. . 3 ⊢ (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	42, 50	spcev 3502	. 2 ⊢ ((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	7, 34, 51	mp2an 682	1 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599   = wceq 1601  ∃wex 1823   ∈ wcel 2107  ∃wrex 3091  Vcvv 3398  ∅c0 4141   ↦ cmpt 4967  dom cdm 5357  ran crn 5358   ↾ cres 5359  suc csuc 5980  Fun wfun 6131   Fn wfn 6132  ‘cfv 6137  ωcom 7345  reccrdg 7790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-om 7346  df-wrecs 7691  df-recs 7753  df-rdg 7791

This theorem is referenced by:  axinf  8840