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Theorem peano5nni 8746
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 variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 1re 7788 . . . 4 1 ∈ ℝ
2 elin 3263 . . . . 5 (1 ∈ (𝐴 ∩ ℝ) ↔ (1 ∈ 𝐴 ∧ 1 ∈ ℝ))
32biimpri 132 . . . 4 ((1 ∈ 𝐴 ∧ 1 ∈ ℝ) → 1 ∈ (𝐴 ∩ ℝ))
41, 3mpan2 422 . . 3 (1 ∈ 𝐴 → 1 ∈ (𝐴 ∩ ℝ))
5 inss1 3300 . . . . 5 (𝐴 ∩ ℝ) ⊆ 𝐴
6 ssralv 3165 . . . . 5 ((𝐴 ∩ ℝ) ⊆ 𝐴 → (∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ 𝐴))
75, 6ax-mp 5 . . . 4 (∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ 𝐴)
8 inss2 3301 . . . . . . . 8 (𝐴 ∩ ℝ) ⊆ ℝ
98sseli 3097 . . . . . . 7 (𝑥 ∈ (𝐴 ∩ ℝ) → 𝑥 ∈ ℝ)
10 1red 7804 . . . . . . 7 (𝑥 ∈ (𝐴 ∩ ℝ) → 1 ∈ ℝ)
119, 10readdcld 7818 . . . . . 6 (𝑥 ∈ (𝐴 ∩ ℝ) → (𝑥 + 1) ∈ ℝ)
12 elin 3263 . . . . . . 7 ((𝑥 + 1) ∈ (𝐴 ∩ ℝ) ↔ ((𝑥 + 1) ∈ 𝐴 ∧ (𝑥 + 1) ∈ ℝ))
1312simplbi2com 1421 . . . . . 6 ((𝑥 + 1) ∈ ℝ → ((𝑥 + 1) ∈ 𝐴 → (𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
1411, 13syl 14 . . . . 5 (𝑥 ∈ (𝐴 ∩ ℝ) → ((𝑥 + 1) ∈ 𝐴 → (𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
1514ralimia 2496 . . . 4 (∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ))
167, 15syl 14 . . 3 (∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ))
17 reex 7777 . . . . 5 ℝ ∈ V
1817inex2 4070 . . . 4 (𝐴 ∩ ℝ) ∈ V
19 eleq2 2204 . . . . . . 7 (𝑦 = (𝐴 ∩ ℝ) → (1 ∈ 𝑦 ↔ 1 ∈ (𝐴 ∩ ℝ)))
20 eleq2 2204 . . . . . . . 8 (𝑦 = (𝐴 ∩ ℝ) → ((𝑥 + 1) ∈ 𝑦 ↔ (𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
2120raleqbi1dv 2637 . . . . . . 7 (𝑦 = (𝐴 ∩ ℝ) → (∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
2219, 21anbi12d 465 . . . . . 6 (𝑦 = (𝐴 ∩ ℝ) → ((1 ∈ 𝑦 ∧ ∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦) ↔ (1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ))))
2322elabg 2833 . . . . 5 ((𝐴 ∩ ℝ) ∈ V → ((𝐴 ∩ ℝ) ∈ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦)} ↔ (1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ))))
24 dfnn2 8745 . . . . . 6 ℕ = {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦)}
25 intss1 3793 . . . . . 6 ((𝐴 ∩ ℝ) ∈ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦)} → {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦)} ⊆ (𝐴 ∩ ℝ))
2624, 25eqsstrid 3147 . . . . 5 ((𝐴 ∩ ℝ) ∈ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦)} → ℕ ⊆ (𝐴 ∩ ℝ))
2723, 26syl6bir 163 . . . 4 ((𝐴 ∩ ℝ) ∈ V → ((1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)) → ℕ ⊆ (𝐴 ∩ ℝ)))
2818, 27ax-mp 5 . . 3 ((1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)) → ℕ ⊆ (𝐴 ∩ ℝ))
294, 16, 28syl2an 287 . 2 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ (𝐴 ∩ ℝ))
3029, 5sstrdi 3113 1 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  {cab 2126  wral 2417  Vcvv 2689  cin 3074  wss 3075   cint 3778  (class class class)co 5781  cr 7642  1c1 7644   + caddc 7646  cn 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-cnex 7734  ax-resscn 7735  ax-1re 7737  ax-addrcl 7740
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-in 3081  df-ss 3088  df-int 3779  df-inn 8744
This theorem is referenced by:  nnssre  8747  nnind  8759
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