Theorem peano5nni 7917

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.

Step	Hyp	Ref	Expression
1		1re 7026	. . . 4 ⊢ 1 ∈ ℝ
2		elin 3126	. . . . 5 ⊢ (1 ∈ (𝐴 ∩ ℝ) ↔ (1 ∈ 𝐴 ∧ 1 ∈ ℝ))
3	2	biimpri 124	. . . 4 ⊢ ((1 ∈ 𝐴 ∧ 1 ∈ ℝ) → 1 ∈ (𝐴 ∩ ℝ))
4	1, 3	mpan2 401	. . 3 ⊢ (1 ∈ 𝐴 → 1 ∈ (𝐴 ∩ ℝ))
5		inss1 3157	. . . . 5 ⊢ (𝐴 ∩ ℝ) ⊆ 𝐴
6		ssralv 3004	. . . . 5 ⊢ ((𝐴 ∩ ℝ) ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ 𝐴))
7	5, 6	ax-mp 7	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ 𝐴)
8		inss2 3158	. . . . . . . 8 ⊢ (𝐴 ∩ ℝ) ⊆ ℝ
9	8	sseli 2941	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → 𝑥 ∈ ℝ)
10		1red 7042	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → 1 ∈ ℝ)
11	9, 10	readdcld 7055	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → (𝑥 + 1) ∈ ℝ)
12		elin 3126	. . . . . . 7 ⊢ ((𝑥 + 1) ∈ (𝐴 ∩ ℝ) ↔ ((𝑥 + 1) ∈ 𝐴 ∧ (𝑥 + 1) ∈ ℝ))
13	12	simplbi2com 1333	. . . . . 6 ⊢ ((𝑥 + 1) ∈ ℝ → ((𝑥 + 1) ∈ 𝐴 → (𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
14	11, 13	syl 14	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → ((𝑥 + 1) ∈ 𝐴 → (𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
15	14	ralimia 2382	. . . 4 ⊢ (∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ))
16	7, 15	syl 14	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ))
17		reex 7015	. . . . 5 ⊢ ℝ ∈ V
18	17	inex2 3892	. . . 4 ⊢ (𝐴 ∩ ℝ) ∈ V
19		eleq2 2101	. . . . . . 7 ⊢ (𝑦 = (𝐴 ∩ ℝ) → (1 ∈ 𝑦 ↔ 1 ∈ (𝐴 ∩ ℝ)))
20		eleq2 2101	. . . . . . . 8 ⊢ (𝑦 = (𝐴 ∩ ℝ) → ((𝑥 + 1) ∈ 𝑦 ↔ (𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
21	20	raleqbi1dv 2513	. . . . . . 7 ⊢ (𝑦 = (𝐴 ∩ ℝ) → (∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
22	19, 21	anbi12d 442	. . . . . 6 ⊢ (𝑦 = (𝐴 ∩ ℝ) → ((1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦) ↔ (1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ))))
23	22	elabg 2688	. . . . 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 7916	. . . . . 6 ⊢ ℕ = ∩ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)}
25		intss1 3630	. . . . . 6 ⊢ ((𝐴 ∩ ℝ) ∈ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)} → ∩ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)} ⊆ (𝐴 ∩ ℝ))
26	24, 25	syl5eqss 2989	. . . . 5 ⊢ ((𝐴 ∩ ℝ) ∈ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)} → ℕ ⊆ (𝐴 ∩ ℝ))
27	23, 26	syl6bir 153	. . . 4 ⊢ ((𝐴 ∩ ℝ) ∈ V → ((1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)) → ℕ ⊆ (𝐴 ∩ ℝ)))
28	18, 27	ax-mp 7	. . 3 ⊢ ((1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)) → ℕ ⊆ (𝐴 ∩ ℝ))
29	4, 16, 28	syl2an 273	. 2 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ (𝐴 ∩ ℝ))
30	29, 5	syl6ss 2957	1 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  Vcvv 2557   ∩ cin 2916   ⊆ wss 2917  ∩ cint 3615  (class class class)co 5512  ℝcr 6888  1c1 6890   + caddc 6892  ℕcn 7914

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-cnex 6975  ax-resscn 6976  ax-1re 6978  ax-addrcl 6981

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-int 3616  df-inn 7915

This theorem is referenced by:  nnssre  7918  nnind  7930