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Theorem peano5nni 12164
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 ∧ ∀𝑥𝐴 (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.
StepHypRef Expression
1 df-nn 12162 . . 3 ℕ = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) “ ω)
2 df-ima 5650 . . 3 (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) “ ω) = ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
31, 2eqtri 2761 . 2 ℕ = ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
4 frfnom 8385 . . . . 5 (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) Fn ω
54a1i 11 . . . 4 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) Fn ω)
6 fveq2 6846 . . . . . . . 8 (𝑦 = ∅ → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅))
76eleq1d 2819 . . . . . . 7 (𝑦 = ∅ → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) ∈ 𝐴))
8 fveq2 6846 . . . . . . . 8 (𝑦 = 𝑧 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧))
98eleq1d 2819 . . . . . . 7 (𝑦 = 𝑧 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴 ↔ ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴))
10 fveq2 6846 . . . . . . . 8 (𝑦 = suc 𝑧 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧))
1110eleq1d 2819 . . . . . . 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 11117 . . . . . . . . 9 1 ∈ ℂ
13 fr0g 8386 . . . . . . . . 9 (1 ∈ ℂ → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) = 1)
1412, 13ax-mp 5 . . . . . . . 8 ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) = 1
15 simpl 484 . . . . . . . 8 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → 1 ∈ 𝐴)
1614, 15eqeltrid 2838 . . . . . . 7 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘∅) ∈ 𝐴)
17 oveq1 7368 . . . . . . . . . . . 12 (𝑥 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → (𝑥 + 1) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
1817eleq1d 2819 . . . . . . . . . . 11 (𝑥 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → ((𝑥 + 1) ∈ 𝐴 ↔ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
1918rspccv 3580 . . . . . . . . . 10 (∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
2019ad2antlr 726 . . . . . . . . 9 (((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (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 7394 . . . . . . . . . . . 12 (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ V
22 eqid 2733 . . . . . . . . . . . . 13 (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
23 oveq1 7368 . . . . . . . . . . . . 13 (𝑦 = 𝑛 → (𝑦 + 1) = (𝑛 + 1))
24 oveq1 7368 . . . . . . . . . . . . 13 (𝑦 = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) → (𝑦 + 1) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
2522, 23, 24frsucmpt2 8390 . . . . . . . . . . . 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))
2621, 25mpan2 690 . . . . . . . . . . 11 (𝑧 ∈ ω → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1))
2726eleq1d 2819 . . . . . . . . . 10 (𝑧 ∈ ω → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴 ↔ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) + 1) ∈ 𝐴))
2827adantl 483 . . . . . . . . 9 (((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (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))
2920, 28sylibrd 259 . . . . . . . 8 (((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) ∧ 𝑧 ∈ ω) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴))
3029expcom 415 . . . . . . 7 (𝑧 ∈ ω → ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑧) ∈ 𝐴 → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘suc 𝑧) ∈ 𝐴)))
317, 9, 11, 16, 30finds2 7841 . . . . . 6 (𝑦 ∈ ω → ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
3231com12 32 . . . . 5 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → (𝑦 ∈ ω → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴))
3332ralrimiv 3139 . . . 4 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ∀𝑦 ∈ ω ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)‘𝑦) ∈ 𝐴)
34 ffnfv 7070 . . . 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))
355, 33, 34sylanbrc 584 . . 3 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω):ω⟶𝐴)
3635frnd 6680 . 2 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ran (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω) ⊆ 𝐴)
373, 36eqsstrid 3996 1 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3447  wss 3914  c0 4286  cmpt 5192  ran crn 5638  cres 5639  cima 5640  suc csuc 6323   Fn wfn 6495  wf 6496  cfv 6500  (class class class)co 7361  ωcom 7806  reccrdg 8359  cc 11057  1c1 11060   + caddc 11062  cn 12161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676  ax-1cn 11117
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-nn 12162
This theorem is referenced by:  nnssre  12165  nnsscn  12166  dfnn2  12174  nnind  12179  nnindf  31771
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