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Theorem peano5nni 8397
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 7466 . . . 4 1 ∈ ℝ
2 elin 3181 . . . . 5 (1 ∈ (𝐴 ∩ ℝ) ↔ (1 ∈ 𝐴 ∧ 1 ∈ ℝ))
32biimpri 131 . . . 4 ((1 ∈ 𝐴 ∧ 1 ∈ ℝ) → 1 ∈ (𝐴 ∩ ℝ))
41, 3mpan2 416 . . 3 (1 ∈ 𝐴 → 1 ∈ (𝐴 ∩ ℝ))
5 inss1 3218 . . . . 5 (𝐴 ∩ ℝ) ⊆ 𝐴
6 ssralv 3083 . . . . 5 ((𝐴 ∩ ℝ) ⊆ 𝐴 → (∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ 𝐴))
75, 6ax-mp 7 . . . 4 (∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ 𝐴)
8 inss2 3219 . . . . . . . 8 (𝐴 ∩ ℝ) ⊆ ℝ
98sseli 3019 . . . . . . 7 (𝑥 ∈ (𝐴 ∩ ℝ) → 𝑥 ∈ ℝ)
10 1red 7482 . . . . . . 7 (𝑥 ∈ (𝐴 ∩ ℝ) → 1 ∈ ℝ)
119, 10readdcld 7496 . . . . . 6 (𝑥 ∈ (𝐴 ∩ ℝ) → (𝑥 + 1) ∈ ℝ)
12 elin 3181 . . . . . . 7 ((𝑥 + 1) ∈ (𝐴 ∩ ℝ) ↔ ((𝑥 + 1) ∈ 𝐴 ∧ (𝑥 + 1) ∈ ℝ))
1312simplbi2com 1378 . . . . . 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 2436 . . . 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 7455 . . . . 5 ℝ ∈ V
1817inex2 3966 . . . 4 (𝐴 ∩ ℝ) ∈ V
19 eleq2 2151 . . . . . . 7 (𝑦 = (𝐴 ∩ ℝ) → (1 ∈ 𝑦 ↔ 1 ∈ (𝐴 ∩ ℝ)))
20 eleq2 2151 . . . . . . . 8 (𝑦 = (𝐴 ∩ ℝ) → ((𝑥 + 1) ∈ 𝑦 ↔ (𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
2120raleqbi1dv 2570 . . . . . . 7 (𝑦 = (𝐴 ∩ ℝ) → (∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
2219, 21anbi12d 457 . . . . . 6 (𝑦 = (𝐴 ∩ ℝ) → ((1 ∈ 𝑦 ∧ ∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦) ↔ (1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ))))
2322elabg 2759 . . . . 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 8396 . . . . . 6 ℕ = {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦)}
25 intss1 3698 . . . . . 6 ((𝐴 ∩ ℝ) ∈ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦)} → {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦)} ⊆ (𝐴 ∩ ℝ))
2624, 25syl5eqss 3068 . . . . 5 ((𝐴 ∩ ℝ) ∈ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥𝑦 (𝑥 + 1) ∈ 𝑦)} → ℕ ⊆ (𝐴 ∩ ℝ))
2723, 26syl6bir 162 . . . 4 ((𝐴 ∩ ℝ) ∈ V → ((1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)) → ℕ ⊆ (𝐴 ∩ ℝ)))
2818, 27ax-mp 7 . . 3 ((1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)) → ℕ ⊆ (𝐴 ∩ ℝ))
294, 16, 28syl2an 283 . 2 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ (𝐴 ∩ ℝ))
3029, 5syl6ss 3035 1 ((1 ∈ 𝐴 ∧ ∀𝑥𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  {cab 2074  wral 2359  Vcvv 2619  cin 2996  wss 2997   cint 3683  (class class class)co 5634  cr 7328  1c1 7330   + caddc 7332  cn 8394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-cnex 7415  ax-resscn 7416  ax-1re 7418  ax-addrcl 7421
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3003  df-ss 3010  df-int 3684  df-inn 8395
This theorem is referenced by:  nnssre  8398  nnind  8410
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