Theorem peano5 7896

Description: The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43, except our proof does not require the Axiom of Infinity. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as Theorem findes 7903. (Contributed by NM, 18-Feb-2004.) Avoid ax-10 2179, ax-12 2216. (Revised by GG, 3-Oct-2024.)

Assertion

Ref	Expression
peano5	⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem peano5

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifn 4086	. . . . . 6 ⊢ (𝑧 ∈ (ω ∖ 𝐴) → ¬ 𝑧 ∈ 𝐴)
2	1	adantl 487	. . . . 5 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑧 ∈ (ω ∖ 𝐴)) → ¬ 𝑧 ∈ 𝐴)
3		eldifi 4085	. . . . . . . 8 ⊢ (𝑧 ∈ (ω ∖ 𝐴) → 𝑧 ∈ ω)
4		elndif 4087	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 → ¬ ∅ ∈ (ω ∖ 𝐴))
5		eleq1 2853	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑧 ∈ (ω ∖ 𝐴) ↔ ∅ ∈ (ω ∖ 𝐴)))
6	5	biimpcd 252	. . . . . . . . . 10 ⊢ (𝑧 ∈ (ω ∖ 𝐴) → (𝑧 = ∅ → ∅ ∈ (ω ∖ 𝐴)))
7	6	necon3bd 2974	. . . . . . . . 9 ⊢ (𝑧 ∈ (ω ∖ 𝐴) → (¬ ∅ ∈ (ω ∖ 𝐴) → 𝑧 ≠ ∅))
8	4, 7	mpan9 516	. . . . . . . 8 ⊢ ((∅ ∈ 𝐴 ∧ 𝑧 ∈ (ω ∖ 𝐴)) → 𝑧 ≠ ∅)
9		nnsuc 7886	. . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ 𝑧 ≠ ∅) → ∃𝑦 ∈ ω 𝑧 = suc 𝑦)
10	3, 8, 9	syl2an2 699	. . . . . . 7 ⊢ ((∅ ∈ 𝐴 ∧ 𝑧 ∈ (ω ∖ 𝐴)) → ∃𝑦 ∈ ω 𝑧 = suc 𝑦)
11	10	ad4ant13 764	. . . . . 6 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑧 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → ∃𝑦 ∈ ω 𝑧 = suc 𝑦)
12		eleq1w 2848	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
13		suceq 6433	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
14	13	eleq1d 2850	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
15	12, 14	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 → suc 𝑦 ∈ 𝐴)))
16	15	rspccv 3580	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑦 ∈ ω → (𝑦 ∈ 𝐴 → suc 𝑦 ∈ 𝐴)))
17		vex 3461	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
18	17	sucid 6449	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ suc 𝑦
19		eleq2 2854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = suc 𝑦 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ suc 𝑦))
20	18, 19	mpbiri 261	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑦 → 𝑦 ∈ 𝑧)
21		eleq1 2853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ ω ↔ suc 𝑦 ∈ ω))
22		peano2b 7885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
23	21, 22	bitr4di 292	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ ω ↔ 𝑦 ∈ ω))
24		minel 4426	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑧 ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → ¬ 𝑦 ∈ (ω ∖ 𝐴))
25		neldif 4088	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ω ∧ ¬ 𝑦 ∈ (ω ∖ 𝐴)) → 𝑦 ∈ 𝐴)
26	24, 25	sylan2 605	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ω ∧ (𝑦 ∈ 𝑧 ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅)) → 𝑦 ∈ 𝐴)
27	26	exp32 426	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ω → (𝑦 ∈ 𝑧 → (((ω ∖ 𝐴) ∩ 𝑧) = ∅ → 𝑦 ∈ 𝐴)))
28	23, 27	biimtrdi 256	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ ω → (𝑦 ∈ 𝑧 → (((ω ∖ 𝐴) ∩ 𝑧) = ∅ → 𝑦 ∈ 𝐴))))
29	20, 28	mpid 45	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ ω → (((ω ∖ 𝐴) ∩ 𝑧) = ∅ → 𝑦 ∈ 𝐴)))
30	3, 29	syl5 35	. . . . . . . . . . . . . 14 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑧) = ∅ → 𝑦 ∈ 𝐴)))
31	30	impd 416	. . . . . . . . . . . . 13 ⊢ (𝑧 = suc 𝑦 → ((𝑧 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → 𝑦 ∈ 𝐴))
32		eleq1a 2860	. . . . . . . . . . . . . 14 ⊢ (suc 𝑦 ∈ 𝐴 → (𝑧 = suc 𝑦 → 𝑧 ∈ 𝐴))
33	32	com12 33	. . . . . . . . . . . . 13 ⊢ (𝑧 = suc 𝑦 → (suc 𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴))
34	31, 33	imim12d 82	. . . . . . . . . . . 12 ⊢ (𝑧 = suc 𝑦 → ((𝑦 ∈ 𝐴 → suc 𝑦 ∈ 𝐴) → ((𝑧 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → 𝑧 ∈ 𝐴)))
35	34	com13 89	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → ((𝑦 ∈ 𝐴 → suc 𝑦 ∈ 𝐴) → (𝑧 = suc 𝑦 → 𝑧 ∈ 𝐴)))
36	16, 35	sylan9 517	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅)) → (𝑦 ∈ ω → (𝑧 = suc 𝑦 → 𝑧 ∈ 𝐴)))
37	36	rexlimdv 3166	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅)) → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → 𝑧 ∈ 𝐴))
38	37	exp32 426	. . . . . . . 8 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑧 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑧) = ∅ → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → 𝑧 ∈ 𝐴))))
39	38	a1i 11	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑧 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑧) = ∅ → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → 𝑧 ∈ 𝐴)))))
40	39	imp41 431	. . . . . 6 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑧 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → 𝑧 ∈ 𝐴))
41	11, 40	mpd 16	. . . . 5 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑧 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → 𝑧 ∈ 𝐴)
42	2, 41	mtand 828	. . . 4 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑧 ∈ (ω ∖ 𝐴)) → ¬ ((ω ∖ 𝐴) ∩ 𝑧) = ∅)
43	42	nrexdv 3162	. . 3 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ¬ ∃𝑧 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑧) = ∅)
44		ordom 7878	. . . . 5 ⊢ Ord ω
45		difss 4090	. . . . 5 ⊢ (ω ∖ 𝐴) ⊆ ω
46		tz7.5 6385	. . . . 5 ⊢ ((Ord ω ∧ (ω ∖ 𝐴) ⊆ ω ∧ (ω ∖ 𝐴) ≠ ∅) → ∃𝑧 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑧) = ∅)
47	44, 45, 46	mp3an12 1480	. . . 4 ⊢ ((ω ∖ 𝐴) ≠ ∅ → ∃𝑧 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑧) = ∅)
48	47	necon1bi 2988	. . 3 ⊢ (¬ ∃𝑧 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑧) = ∅ → (ω ∖ 𝐴) = ∅)
49	43, 48	syl 18	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (ω ∖ 𝐴) = ∅)
50		ssdif0 4321	. 2 ⊢ (ω ⊆ 𝐴 ↔ (ω ∖ 𝐴) = ∅)
51	49, 50	sylibr 237	1 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  Ord word 6363  suc csuc 6366  ωcom 7868

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-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-om 7869

This theorem is used by:  find  7898  finds  7899  finds2  7901  omex  9619  dfom3  9623