Theorem peano5 7872

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 7879. (Contributed by NM, 18-Feb-2004.) Avoid ax-10 2142, ax-12 2178. (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 4098	. . . . . 6 ⊢ (𝑧 ∈ (ω ∖ 𝐴) → ¬ 𝑧 ∈ 𝐴)
2	1	adantl 481	. . . . 5 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑧 ∈ (ω ∖ 𝐴)) → ¬ 𝑧 ∈ 𝐴)
3		eldifi 4097	. . . . . . . 8 ⊢ (𝑧 ∈ (ω ∖ 𝐴) → 𝑧 ∈ ω)
4		elndif 4099	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 → ¬ ∅ ∈ (ω ∖ 𝐴))
5		eleq1 2817	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑧 ∈ (ω ∖ 𝐴) ↔ ∅ ∈ (ω ∖ 𝐴)))
6	5	biimpcd 249	. . . . . . . . . 10 ⊢ (𝑧 ∈ (ω ∖ 𝐴) → (𝑧 = ∅ → ∅ ∈ (ω ∖ 𝐴)))
7	6	necon3bd 2940	. . . . . . . . 9 ⊢ (𝑧 ∈ (ω ∖ 𝐴) → (¬ ∅ ∈ (ω ∖ 𝐴) → 𝑧 ≠ ∅))
8	4, 7	mpan9 506	. . . . . . . 8 ⊢ ((∅ ∈ 𝐴 ∧ 𝑧 ∈ (ω ∖ 𝐴)) → 𝑧 ≠ ∅)
9		nnsuc 7863	. . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ 𝑧 ≠ ∅) → ∃𝑦 ∈ ω 𝑧 = suc 𝑦)
10	3, 8, 9	syl2an2 686	. . . . . . 7 ⊢ ((∅ ∈ 𝐴 ∧ 𝑧 ∈ (ω ∖ 𝐴)) → ∃𝑦 ∈ ω 𝑧 = suc 𝑦)
11	10	ad4ant13 751	. . . . . 6 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑧 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → ∃𝑦 ∈ ω 𝑧 = suc 𝑦)
12		eleq1w 2812	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
13		suceq 6403	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
14	13	eleq1d 2814	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
15	12, 14	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 → suc 𝑦 ∈ 𝐴)))
16	15	rspccv 3588	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑦 ∈ ω → (𝑦 ∈ 𝐴 → suc 𝑦 ∈ 𝐴)))
17		vex 3454	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
18	17	sucid 6419	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ suc 𝑦
19		eleq2 2818	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = suc 𝑦 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ suc 𝑦))
20	18, 19	mpbiri 258	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑦 → 𝑦 ∈ 𝑧)
21		eleq1 2817	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ ω ↔ suc 𝑦 ∈ ω))
22		peano2b 7862	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
23	21, 22	bitr4di 289	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ ω ↔ 𝑦 ∈ ω))
24		minel 4432	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑧 ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → ¬ 𝑦 ∈ (ω ∖ 𝐴))
25		neldif 4100	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ω ∧ ¬ 𝑦 ∈ (ω ∖ 𝐴)) → 𝑦 ∈ 𝐴)
26	24, 25	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ω ∧ (𝑦 ∈ 𝑧 ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅)) → 𝑦 ∈ 𝐴)
27	26	exp32 420	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ω → (𝑦 ∈ 𝑧 → (((ω ∖ 𝐴) ∩ 𝑧) = ∅ → 𝑦 ∈ 𝐴)))
28	23, 27	biimtrdi 253	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ ω → (𝑦 ∈ 𝑧 → (((ω ∖ 𝐴) ∩ 𝑧) = ∅ → 𝑦 ∈ 𝐴))))
29	20, 28	mpid 44	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ ω → (((ω ∖ 𝐴) ∩ 𝑧) = ∅ → 𝑦 ∈ 𝐴)))
30	3, 29	syl5 34	. . . . . . . . . . . . . 14 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑧) = ∅ → 𝑦 ∈ 𝐴)))
31	30	impd 410	. . . . . . . . . . . . 13 ⊢ (𝑧 = suc 𝑦 → ((𝑧 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → 𝑦 ∈ 𝐴))
32		eleq1a 2824	. . . . . . . . . . . . . 14 ⊢ (suc 𝑦 ∈ 𝐴 → (𝑧 = suc 𝑦 → 𝑧 ∈ 𝐴))
33	32	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑧 = suc 𝑦 → (suc 𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴))
34	31, 33	imim12d 81	. . . . . . . . . . . 12 ⊢ (𝑧 = suc 𝑦 → ((𝑦 ∈ 𝐴 → suc 𝑦 ∈ 𝐴) → ((𝑧 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → 𝑧 ∈ 𝐴)))
35	34	com13 88	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → ((𝑦 ∈ 𝐴 → suc 𝑦 ∈ 𝐴) → (𝑧 = suc 𝑦 → 𝑧 ∈ 𝐴)))
36	16, 35	sylan9 507	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅)) → (𝑦 ∈ ω → (𝑧 = suc 𝑦 → 𝑧 ∈ 𝐴)))
37	36	rexlimdv 3133	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑧 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅)) → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → 𝑧 ∈ 𝐴))
38	37	exp32 420	. . . . . . . 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 425	. . . . . 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 15	. . . . 5 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑧 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑧) = ∅) → 𝑧 ∈ 𝐴)
42	2, 41	mtand 815	. . . 4 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑧 ∈ (ω ∖ 𝐴)) → ¬ ((ω ∖ 𝐴) ∩ 𝑧) = ∅)
43	42	nrexdv 3129	. . 3 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ¬ ∃𝑧 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑧) = ∅)
44		ordom 7855	. . . . 5 ⊢ Ord ω
45		difss 4102	. . . . 5 ⊢ (ω ∖ 𝐴) ⊆ ω
46		tz7.5 6356	. . . . 5 ⊢ ((Ord ω ∧ (ω ∖ 𝐴) ⊆ ω ∧ (ω ∖ 𝐴) ≠ ∅) → ∃𝑧 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑧) = ∅)
47	44, 45, 46	mp3an12 1453	. . . 4 ⊢ ((ω ∖ 𝐴) ≠ ∅ → ∃𝑧 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑧) = ∅)
48	47	necon1bi 2954	. . 3 ⊢ (¬ ∃𝑧 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑧) = ∅ → (ω ∖ 𝐴) = ∅)
49	43, 48	syl 17	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (ω ∖ 𝐴) = ∅)
50		ssdif0 4332	. 2 ⊢ (ω ⊆ 𝐴 ↔ (ω ∖ 𝐴) = ∅)
51	49, 50	sylibr 234	1 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054   ∖ cdif 3914   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  Ord word 6334  suc csuc 6337  ωcom 7845

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-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-om 7846

This theorem is referenced by:  find  7874  finds  7875  finds2  7877  omex  9603  dfom3  9607