Theorem peano5nni 11833

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 11831	. . 3 ⊢ ℕ = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) “ ω)
2		df-ima 5564	. . 3 ⊢ (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) “ ω) = ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
3	1, 2	eqtri 2765	. 2 ⊢ ℕ = ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
4		frfnom 8170	. . . . 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 6717	. . . . . . . 8 ⊢ (𝑦 = ∅ → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅))
7	6	eleq1d 2822	. . . . . . 7 ⊢ (𝑦 = ∅ → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) ∈ 𝐴))
8		fveq2 6717	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧))
9	8	eleq1d 2822	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴))
10		fveq2 6717	. . . . . . . 8 ⊢ (𝑦 = suc 𝑧 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧))
11	10	eleq1d 2822	. . . . . . 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 10787	. . . . . . . . 9 ⊢ 1 ∈ ℂ
13		fr0g 8171	. . . . . . . . 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 486	. . . . . . . 8 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → 1 ∈ 𝐴)
16	14, 15	eqeltrid 2842	. . . . . . 7 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) ∈ 𝐴)
17		oveq1 7220	. . . . . . . . . . . 12 ⊢ (𝑥 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → (𝑥 + 1) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
18	17	eleq1d 2822	. . . . . . . . . . 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 3534	. . . . . . . . . 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 727	. . . . . . . . 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 7246	. . . . . . . . . . . 12 ⊢ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ V
22		eqid 2737	. . . . . . . . . . . . 13 ⊢ (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
23		oveq1 7220	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑛 → (𝑦 + 1) = (𝑛 + 1))
24		oveq1 7220	. . . . . . . . . . . . 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 8176	. . . . . . . . . . . 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 691	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
27	26	eleq1d 2822	. . . . . . . . . 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 485	. . . . . . . . 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 417	. . . . . . 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 7678	. . . . . 6 ⊢ (𝑦 ∈ ω → ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
32	31	com12 32	. . . . 5 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → (𝑦 ∈ ω → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
33	32	ralrimiv 3104	. . . 4 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ∀𝑦 ∈ ω ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴)
34		ffnfv 6935	. . . 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 586	. . 3 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω):ω⟶𝐴)
36	35	frnd 6553	. 2 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) ⊆ 𝐴)
37	3, 36	eqsstrid 3949	1 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061  Vcvv 3408   ⊆ wss 3866  ∅c0 4237   ↦ cmpt 5135  ran crn 5552   ↾ cres 5553   “ cima 5554  suc csuc 6215   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213  ωcom 7644  reccrdg 8145  ℂcc 10727  1c1 10730   + caddc 10732  ℕcn 11830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523  ax-1cn 10787

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-nn 11831

This theorem is referenced by:  nnssre  11834  nnsscn  11835  dfnn2  11843  nnind  11848  nnindf  30853