Theorem peano5nni 8881

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 7919	. . . 4 ⊢ 1 ∈ ℝ
2		elin 3310	. . . . 5 ⊢ (1 ∈ (𝐴 ∩ ℝ) ↔ (1 ∈ 𝐴 ∧ 1 ∈ ℝ))
3	2	biimpri 132	. . . 4 ⊢ ((1 ∈ 𝐴 ∧ 1 ∈ ℝ) → 1 ∈ (𝐴 ∩ ℝ))
4	1, 3	mpan2 423	. . 3 ⊢ (1 ∈ 𝐴 → 1 ∈ (𝐴 ∩ ℝ))
5		inss1 3347	. . . . 5 ⊢ (𝐴 ∩ ℝ) ⊆ 𝐴
6		ssralv 3211	. . . . 5 ⊢ ((𝐴 ∩ ℝ) ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ 𝐴))
7	5, 6	ax-mp 5	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ 𝐴)
8		inss2 3348	. . . . . . . 8 ⊢ (𝐴 ∩ ℝ) ⊆ ℝ
9	8	sseli 3143	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → 𝑥 ∈ ℝ)
10		1red 7935	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → 1 ∈ ℝ)
11	9, 10	readdcld 7949	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → (𝑥 + 1) ∈ ℝ)
12		elin 3310	. . . . . . 7 ⊢ ((𝑥 + 1) ∈ (𝐴 ∩ ℝ) ↔ ((𝑥 + 1) ∈ 𝐴 ∧ (𝑥 + 1) ∈ ℝ))
13	12	simplbi2com 1437	. . . . . 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 2531	. . . 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 7908	. . . . 5 ⊢ ℝ ∈ V
18	17	inex2 4124	. . . 4 ⊢ (𝐴 ∩ ℝ) ∈ V
19		eleq2 2234	. . . . . . 7 ⊢ (𝑦 = (𝐴 ∩ ℝ) → (1 ∈ 𝑦 ↔ 1 ∈ (𝐴 ∩ ℝ)))
20		eleq2 2234	. . . . . . . 8 ⊢ (𝑦 = (𝐴 ∩ ℝ) → ((𝑥 + 1) ∈ 𝑦 ↔ (𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
21	20	raleqbi1dv 2673	. . . . . . 7 ⊢ (𝑦 = (𝐴 ∩ ℝ) → (∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
22	19, 21	anbi12d 470	. . . . . 6 ⊢ (𝑦 = (𝐴 ∩ ℝ) → ((1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦) ↔ (1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ))))
23	22	elabg 2876	. . . . 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 8880	. . . . . 6 ⊢ ℕ = ∩ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)}
25		intss1 3846	. . . . . 6 ⊢ ((𝐴 ∩ ℝ) ∈ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)} → ∩ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)} ⊆ (𝐴 ∩ ℝ))
26	24, 25	eqsstrid 3193	. . . . 5 ⊢ ((𝐴 ∩ ℝ) ∈ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)} → ℕ ⊆ (𝐴 ∩ ℝ))
27	23, 26	syl6bir 163	. . . 4 ⊢ ((𝐴 ∩ ℝ) ∈ V → ((1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)) → ℕ ⊆ (𝐴 ∩ ℝ)))
28	18, 27	ax-mp 5	. . 3 ⊢ ((1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)) → ℕ ⊆ (𝐴 ∩ ℝ))
29	4, 16, 28	syl2an 287	. 2 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ (𝐴 ∩ ℝ))
30	29, 5	sstrdi 3159	1 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  Vcvv 2730   ∩ cin 3120   ⊆ wss 3121  ∩ cint 3831  (class class class)co 5853  ℝcr 7773  1c1 7775   + caddc 7777  ℕcn 8878

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-int 3832  df-inn 8879

This theorem is referenced by:  nnssre  8882  nnind  8894