Theorem peano5nni 12260

Description: Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
peano5nni	⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem peano5nni

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

Step	Hyp	Ref	Expression
1		df-nn 12258	. . 3 ⊢ ℕ = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) “ ω)
2		df-ima 5668	. . 3 ⊢ (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) “ ω) = ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
3	1, 2	eqtri 2783	. 2 ⊢ ℕ = ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
4		frfnom 8424	. . . . 5 ⊢ (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) Fn ω
5	4	a1i 11	. . . 4 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) Fn ω)
6		fveq2 6878	. . . . . . . 8 ⊢ (𝑦 = ∅ → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅))
7	6	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = ∅ → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) ∈ 𝐴))
8		fveq2 6878	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧))
9	8	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴))
10		fveq2 6878	. . . . . . . 8 ⊢ (𝑦 = suc 𝑧 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧))
11	10	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = suc 𝑧 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴))
12		ax-1cn 11182	. . . . . . . . 9 ⊢ 1 ∈ ℂ
13		fr0g 8425	. . . . . . . . 9 ⊢ (1 ∈ ℂ → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) = 1)
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) = 1
15		simpl 488	. . . . . . . 8 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → 1 ∈ 𝐴)
16	14, 15	eqeltrid 2864	. . . . . . 7 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) ∈ 𝐴)
17		oveq1 7420	. . . . . . . . . . . 12 ⊢ (𝑥 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → (𝑥 + 1) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
18	17	eleq1d 2845	. . . . . . . . . . 11 ⊢ (𝑥 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → ((𝑥 + 1) ∈ 𝐴 ↔ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
19	18	rspccv 3573	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
20	19	ad2antlr 740	. . . . . . . . 9 ⊢ (((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) ∧ 𝑧 ∈ ω) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
21		ovex 7446	. . . . . . . . . . . 12 ⊢ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ V
22		eqid 2760	. . . . . . . . . . . . 13 ⊢ (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
23		oveq1 7420	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑛 → (𝑦 + 1) = (𝑛 + 1))
24		oveq1 7420	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → (𝑦 + 1) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
25	22, 23, 24	frsucmpt2 8429	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω ∧ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ V) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
26	21, 25	mpan2 704	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
27	26	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴 ↔ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
28	27	adantl 487	. . . . . . . . 9 ⊢ (((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) ∧ 𝑧 ∈ ω) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴 ↔ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
29	20, 28	sylibrd 262	. . . . . . . 8 ⊢ (((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) ∧ 𝑧 ∈ ω) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴))
30	29	expcom 419	. . . . . . 7 ⊢ (𝑧 ∈ ω → ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴)))
31	7, 9, 11, 16, 30	finds2 7895	. . . . . 6 ⊢ (𝑦 ∈ ω → ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
32	31	com12 33	. . . . 5 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → (𝑦 ∈ ω → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
33	32	ralrimiv 3153	. . . 4 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ∀𝑦 ∈ ω ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴)
34		ffnfv 7112	. . . 4 ⊢ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω):ω⟶𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) Fn ω ∧ ∀𝑦 ∈ ω ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
35	5, 33, 34	sylanbrc 595	. . 3 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω):ω⟶𝐴)
36	35	frnd 6711	. 2 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) ⊆ 𝐴)
37	3, 36	eqsstrid 3969	1 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3899  ∅c0 4279   ↦ cmpt 5186  ran crn 5656   ↾ cres 5657   “ cima 5658  suc csuc 6359   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  ωcom 7862  reccrdg 8398  ℂcc 11122  1c1 11125   + caddc 11127  ℕcn 12257

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736  ax-1cn 11182

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-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-nn 12258

This theorem is used by:  nnssre  12261  nnsscn  12262  dfnn2  12270  nnind  12275  nnindf  33290