Theorem peano5 6981

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 6988. (Contributed by NM, 18-Feb-2004.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem peano5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifn 3695	. . . . . 6 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → ¬ 𝑦 ∈ 𝐴)
2	1	adantl 481	. . . . 5 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ¬ 𝑦 ∈ 𝐴)
3		eldifi 3694	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → 𝑦 ∈ ω)
4	3	adantl 481	. . . . . . . . 9 ⊢ ((∅ ∈ 𝐴 ∧ 𝑦 ∈ (ω ∖ 𝐴)) → 𝑦 ∈ ω)
5		elndif 3696	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ¬ ∅ ∈ (ω ∖ 𝐴))
6		eleq1 2676	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → (𝑦 ∈ (ω ∖ 𝐴) ↔ ∅ ∈ (ω ∖ 𝐴)))
7	6	biimpcd 238	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → (𝑦 = ∅ → ∅ ∈ (ω ∖ 𝐴)))
8	7	necon3bd 2796	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → (¬ ∅ ∈ (ω ∖ 𝐴) → 𝑦 ≠ ∅))
9	5, 8	mpan9 485	. . . . . . . . 9 ⊢ ((∅ ∈ 𝐴 ∧ 𝑦 ∈ (ω ∖ 𝐴)) → 𝑦 ≠ ∅)
10		nnsuc 6974	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≠ ∅) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
11	4, 9, 10	syl2anc 691	. . . . . . . 8 ⊢ ((∅ ∈ 𝐴 ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
12	11	adantlr 747	. . . . . . 7 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
13	12	adantr 480	. . . . . 6 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
14		nfra1 2925	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)
15		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)
16	14, 15	nfan 1816	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅))
17		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
18		rsp 2913	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
19		vex 3176	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
20	19	sucid 5721	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ suc 𝑥
21		eleq2 2677	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = suc 𝑥 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ suc 𝑥))
22	20, 21	mpbiri 247	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = suc 𝑥 → 𝑥 ∈ 𝑦)
23		eleq1 2676	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω ↔ suc 𝑥 ∈ ω))
24		peano2b 6973	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
25	23, 24	syl6bbr 277	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω ↔ 𝑥 ∈ ω))
26		minel 3985	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝑦 ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ¬ 𝑥 ∈ (ω ∖ 𝐴))
27		neldif 3697	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ω ∧ ¬ 𝑥 ∈ (ω ∖ 𝐴)) → 𝑥 ∈ 𝐴)
28	26, 27	sylan2 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ω ∧ (𝑥 ∈ 𝑦 ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → 𝑥 ∈ 𝐴)
29	28	exp32 629	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ω → (𝑥 ∈ 𝑦 → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴)))
30	25, 29	syl6bi 242	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω → (𝑥 ∈ 𝑦 → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴))))
31	22, 30	mpid 43	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴)))
32	3, 31	syl5 33	. . . . . . . . . . . . . 14 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴)))
33	32	impd 446	. . . . . . . . . . . . 13 ⊢ (𝑦 = suc 𝑥 → ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑥 ∈ 𝐴))
34		eleq1a 2683	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ 𝐴 → (𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))
35	34	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑦 = suc 𝑥 → (suc 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))
36	33, 35	imim12d 79	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑥 → ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑦 ∈ 𝐴)))
37	36	com13 86	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴)))
38	18, 37	sylan9 687	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → (𝑥 ∈ ω → (𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴)))
39	16, 17, 38	rexlimd 3008	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))
40	39	exp32 629	. . . . . . . 8 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))))
41	40	a1i 11	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴)))))
42	41	imp41 617	. . . . . 6 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))
43	13, 42	mpd 15	. . . . 5 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑦 ∈ 𝐴)
44	2, 43	mtand 689	. . . 4 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ¬ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)
45	44	nrexdv 2984	. . 3 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ¬ ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
46		ordom 6966	. . . . 5 ⊢ Ord ω
47		difss 3699	. . . . 5 ⊢ (ω ∖ 𝐴) ⊆ ω
48		tz7.5 5661	. . . . 5 ⊢ ((Ord ω ∧ (ω ∖ 𝐴) ⊆ ω ∧ (ω ∖ 𝐴) ≠ ∅) → ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
49	46, 47, 48	mp3an12 1406	. . . 4 ⊢ ((ω ∖ 𝐴) ≠ ∅ → ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
50	49	necon1bi 2810	. . 3 ⊢ (¬ ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (ω ∖ 𝐴) = ∅)
51	45, 50	syl 17	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (ω ∖ 𝐴) = ∅)
52		ssdif0 3896	. 2 ⊢ (ω ⊆ 𝐴 ↔ (ω ∖ 𝐴) = ∅)
53	51, 52	sylibr 223	1 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  Ord word 5639  suc csuc 5642  ωcom 6957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958

This theorem is referenced by:  find  6983  finds  6984  finds2  6986  omex  8423  dfom3  8427