Theorem peano5nni 8042

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 7118	. . . 4 ⊢ 1 ∈ ℝ
2		elin 3155	. . . . 5 ⊢ (1 ∈ (𝐴 ∩ ℝ) ↔ (1 ∈ 𝐴 ∧ 1 ∈ ℝ))
3	2	biimpri 131	. . . 4 ⊢ ((1 ∈ 𝐴 ∧ 1 ∈ ℝ) → 1 ∈ (𝐴 ∩ ℝ))
4	1, 3	mpan2 415	. . 3 ⊢ (1 ∈ 𝐴 → 1 ∈ (𝐴 ∩ ℝ))
5		inss1 3186	. . . . 5 ⊢ (𝐴 ∩ ℝ) ⊆ 𝐴
6		ssralv 3058	. . . . 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 3187	. . . . . . . 8 ⊢ (𝐴 ∩ ℝ) ⊆ ℝ
9	8	sseli 2995	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → 𝑥 ∈ ℝ)
10		1red 7134	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → 1 ∈ ℝ)
11	9, 10	readdcld 7148	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → (𝑥 + 1) ∈ ℝ)
12		elin 3155	. . . . . . 7 ⊢ ((𝑥 + 1) ∈ (𝐴 ∩ ℝ) ↔ ((𝑥 + 1) ∈ 𝐴 ∧ (𝑥 + 1) ∈ ℝ))
13	12	simplbi2com 1373	. . . . . 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 2424	. . . 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 7107	. . . . 5 ⊢ ℝ ∈ V
18	17	inex2 3913	. . . 4 ⊢ (𝐴 ∩ ℝ) ∈ V
19		eleq2 2142	. . . . . . 7 ⊢ (𝑦 = (𝐴 ∩ ℝ) → (1 ∈ 𝑦 ↔ 1 ∈ (𝐴 ∩ ℝ)))
20		eleq2 2142	. . . . . . . 8 ⊢ (𝑦 = (𝐴 ∩ ℝ) → ((𝑥 + 1) ∈ 𝑦 ↔ (𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
21	20	raleqbi1dv 2557	. . . . . . 7 ⊢ (𝑦 = (𝐴 ∩ ℝ) → (∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
22	19, 21	anbi12d 456	. . . . . 6 ⊢ (𝑦 = (𝐴 ∩ ℝ) → ((1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦) ↔ (1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ))))
23	22	elabg 2739	. . . . 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 8041	. . . . . 6 ⊢ ℕ = ∩ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)}
25		intss1 3651	. . . . . 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 3043	. . . . 5 ⊢ ((𝐴 ∩ ℝ) ∈ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)} → ℕ ⊆ (𝐴 ∩ ℝ))
27	23, 26	syl6bir 162	. . . 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 283	. 2 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ (𝐴 ∩ ℝ))
30	29, 5	syl6ss 3011	1 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  {cab 2067  ∀wral 2348  Vcvv 2601   ∩ cin 2972   ⊆ wss 2973  ∩ cint 3636  (class class class)co 5532  ℝcr 6980  1c1 6982   + caddc 6984  ℕcn 8039

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-cnex 7067  ax-resscn 7068  ax-1re 7070  ax-addrcl 7073

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-int 3637  df-inn 8040

This theorem is referenced by:  nnssre  8043  nnind  8055