Theorem peano5 7367

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 7374. (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 3956	. . . . . 6 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → ¬ 𝑦 ∈ 𝐴)
2	1	adantl 475	. . . . 5 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ¬ 𝑦 ∈ 𝐴)
3		eldifi 3955	. . . . . . . 8 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → 𝑦 ∈ ω)
4		elndif 3957	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 → ¬ ∅ ∈ (ω ∖ 𝐴))
5		eleq1 2847	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∈ (ω ∖ 𝐴) ↔ ∅ ∈ (ω ∖ 𝐴)))
6	5	biimpcd 241	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → (𝑦 = ∅ → ∅ ∈ (ω ∖ 𝐴)))
7	6	necon3bd 2983	. . . . . . . . 9 ⊢ (𝑦 ∈ (ω ∖ 𝐴) → (¬ ∅ ∈ (ω ∖ 𝐴) → 𝑦 ≠ ∅))
8	4, 7	mpan9 502	. . . . . . . 8 ⊢ ((∅ ∈ 𝐴 ∧ 𝑦 ∈ (ω ∖ 𝐴)) → 𝑦 ≠ ∅)
9		nnsuc 7360	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≠ ∅) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
10	3, 8, 9	syl2an2 676	. . . . . . 7 ⊢ ((∅ ∈ 𝐴 ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
11	10	ad4ant13 741	. . . . . 6 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
12		nfra1 3123	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)
13		nfv 1957	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)
14	12, 13	nfan 1946	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅))
15		nfv 1957	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
16		rsp 3111	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
17		vex 3401	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
18	17	sucid 6055	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ suc 𝑥
19		eleq2 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = suc 𝑥 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ suc 𝑥))
20	18, 19	mpbiri 250	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = suc 𝑥 → 𝑥 ∈ 𝑦)
21		eleq1 2847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω ↔ suc 𝑥 ∈ ω))
22		peano2b 7359	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
23	21, 22	syl6bbr 281	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω ↔ 𝑥 ∈ ω))
24		minel 4258	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝑦 ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ¬ 𝑥 ∈ (ω ∖ 𝐴))
25		neldif 3958	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ω ∧ ¬ 𝑥 ∈ (ω ∖ 𝐴)) → 𝑥 ∈ 𝐴)
26	24, 25	sylan2 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ω ∧ (𝑥 ∈ 𝑦 ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → 𝑥 ∈ 𝐴)
27	26	exp32 413	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ω → (𝑥 ∈ 𝑦 → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴)))
28	23, 27	syl6bi 245	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω → (𝑥 ∈ 𝑦 → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴))))
29	20, 28	mpid 44	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ ω → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴)))
30	3, 29	syl5 34	. . . . . . . . . . . . . 14 ⊢ (𝑦 = suc 𝑥 → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → 𝑥 ∈ 𝐴)))
31	30	impd 400	. . . . . . . . . . . . 13 ⊢ (𝑦 = suc 𝑥 → ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑥 ∈ 𝐴))
32		eleq1a 2854	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ 𝐴 → (𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))
33	32	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑦 = suc 𝑥 → (suc 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))
34	31, 33	imim12d 81	. . . . . . . . . . . 12 ⊢ (𝑦 = suc 𝑥 → ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑦 ∈ 𝐴)))
35	34	com13 88	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴)))
36	16, 35	sylan9 503	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → (𝑥 ∈ ω → (𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴)))
37	14, 15, 36	rexlimd 3208	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ (ω ∖ 𝐴) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)) → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))
38	37	exp32 413	. . . . . . . 8 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))))
39	38	a1i 11	. . . . . . 7 ⊢ (∅ ∈ 𝐴 → (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑦 ∈ (ω ∖ 𝐴) → (((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴)))))
40	39	imp41 418	. . . . . 6 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 → 𝑦 ∈ 𝐴))
41	11, 40	mpd 15	. . . . 5 ⊢ ((((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) ∧ ((ω ∖ 𝐴) ∩ 𝑦) = ∅) → 𝑦 ∈ 𝐴)
42	2, 41	mtand 806	. . . 4 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ∧ 𝑦 ∈ (ω ∖ 𝐴)) → ¬ ((ω ∖ 𝐴) ∩ 𝑦) = ∅)
43	42	nrexdv 3182	. . 3 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ¬ ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
44		ordom 7352	. . . . 5 ⊢ Ord ω
45		difss 3960	. . . . 5 ⊢ (ω ∖ 𝐴) ⊆ ω
46		tz7.5 5997	. . . . 5 ⊢ ((Ord ω ∧ (ω ∖ 𝐴) ⊆ ω ∧ (ω ∖ 𝐴) ≠ ∅) → ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
47	44, 45, 46	mp3an12 1524	. . . 4 ⊢ ((ω ∖ 𝐴) ≠ ∅ → ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅)
48	47	necon1bi 2997	. . 3 ⊢ (¬ ∃𝑦 ∈ (ω ∖ 𝐴)((ω ∖ 𝐴) ∩ 𝑦) = ∅ → (ω ∖ 𝐴) = ∅)
49	43, 48	syl 17	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (ω ∖ 𝐴) = ∅)
50		ssdif0 4172	. 2 ⊢ (ω ⊆ 𝐴 ↔ (ω ∖ 𝐴) = ∅)
51	49, 50	sylibr 226	1 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  ∃wrex 3091   ∖ cdif 3789   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  Ord word 5975  suc csuc 5978  ωcom 7343

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-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226

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-rab 3099  df-v 3400  df-sbc 3653  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 4672  df-br 4887  df-opab 4949  df-tr 4988  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-om 7344

This theorem is referenced by:  find  7369  finds  7370  finds2  7372  omex  8837  dfom3  8841