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Theorem peano5nni 7698
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 variable y is distinct from all other variables.
StepHypRef Expression
1 1re 6824 . . . 4 1
2 elin 3120 . . . . 5 (1 (A ∩ ℝ) ↔ (1 A 1 ℝ))
32biimpri 124 . . . 4 ((1 A 1 ℝ) → 1 (A ∩ ℝ))
41, 3mpan2 401 . . 3 (1 A → 1 (A ∩ ℝ))
5 inss1 3151 . . . . 5 (A ∩ ℝ) ⊆ A
6 ssralv 2998 . . . . 5 ((A ∩ ℝ) ⊆ A → (x A (x + 1) Ax (A ∩ ℝ)(x + 1) A))
75, 6ax-mp 7 . . . 4 (x A (x + 1) Ax (A ∩ ℝ)(x + 1) A)
8 inss2 3152 . . . . . . . 8 (A ∩ ℝ) ⊆ ℝ
98sseli 2935 . . . . . . 7 (x (A ∩ ℝ) → x ℝ)
10 1red 6840 . . . . . . 7 (x (A ∩ ℝ) → 1 ℝ)
119, 10readdcld 6852 . . . . . 6 (x (A ∩ ℝ) → (x + 1) ℝ)
12 elin 3120 . . . . . . 7 ((x + 1) (A ∩ ℝ) ↔ ((x + 1) A (x + 1) ℝ))
1312simplbi2com 1330 . . . . . 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 2376 . . . 4 (x (A ∩ ℝ)(x + 1) Ax (A ∩ ℝ)(x + 1) (A ∩ ℝ))
167, 15syl 14 . . 3 (x A (x + 1) Ax (A ∩ ℝ)(x + 1) (A ∩ ℝ))
17 reex 6813 . . . . 5 V
1817inex2 3883 . . . 4 (A ∩ ℝ) V
19 eleq2 2098 . . . . . . 7 (y = (A ∩ ℝ) → (1 y ↔ 1 (A ∩ ℝ)))
20 eleq2 2098 . . . . . . . 8 (y = (A ∩ ℝ) → ((x + 1) y ↔ (x + 1) (A ∩ ℝ)))
2120raleqbi1dv 2507 . . . . . . 7 (y = (A ∩ ℝ) → (x y (x + 1) yx (A ∩ ℝ)(x + 1) (A ∩ ℝ)))
2219, 21anbi12d 442 . . . . . 6 (y = (A ∩ ℝ) → ((1 y x y (x + 1) y) ↔ (1 (A ∩ ℝ) x (A ∩ ℝ)(x + 1) (A ∩ ℝ))))
2322elabg 2682 . . . . 5 ((A ∩ ℝ) V → ((A ∩ ℝ) {y ∣ (1 y x y (x + 1) y)} ↔ (1 (A ∩ ℝ) x (A ∩ ℝ)(x + 1) (A ∩ ℝ))))
24 dfnn2 7697 . . . . . 6 ℕ = {y ∣ (1 y x y (x + 1) y)}
25 intss1 3621 . . . . . 6 ((A ∩ ℝ) {y ∣ (1 y x y (x + 1) y)} → {y ∣ (1 y x y (x + 1) y)} ⊆ (A ∩ ℝ))
2624, 25syl5eqss 2983 . . . . 5 ((A ∩ ℝ) {y ∣ (1 y x y (x + 1) y)} → ℕ ⊆ (A ∩ ℝ))
2723, 26syl6bir 153 . . . 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 273 . 2 ((1 A x A (x + 1) A) → ℕ ⊆ (A ∩ ℝ))
3029, 5syl6ss 2951 1 ((1 A x A (x + 1) A) → ℕ ⊆ A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {cab 2023  wral 2300  Vcvv 2551  cin 2910  wss 2911   cint 3606  (class class class)co 5455  cr 6710  1c1 6712   + caddc 6714  cn 7695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-cnex 6774  ax-resscn 6775  ax-1re 6777  ax-addrcl 6780
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-int 3607  df-inn 7696
This theorem is referenced by:  nnssre  7699  nnind  7711
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