Theorem peano5nni 8921

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 7955	. . . 4 ⊢ 1 ∈ ℝ
2		elin 3318	. . . . 5 ⊢ (1 ∈ (𝐴 ∩ ℝ) ↔ (1 ∈ 𝐴 ∧ 1 ∈ ℝ))
3	2	biimpri 133	. . . 4 ⊢ ((1 ∈ 𝐴 ∧ 1 ∈ ℝ) → 1 ∈ (𝐴 ∩ ℝ))
4	1, 3	mpan2 425	. . 3 ⊢ (1 ∈ 𝐴 → 1 ∈ (𝐴 ∩ ℝ))
5		inss1 3355	. . . . 5 ⊢ (𝐴 ∩ ℝ) ⊆ 𝐴
6		ssralv 3219	. . . . 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 3356	. . . . . . . 8 ⊢ (𝐴 ∩ ℝ) ⊆ ℝ
9	8	sseli 3151	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → 𝑥 ∈ ℝ)
10		1red 7971	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → 1 ∈ ℝ)
11	9, 10	readdcld 7986	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ ℝ) → (𝑥 + 1) ∈ ℝ)
12		elin 3318	. . . . . . 7 ⊢ ((𝑥 + 1) ∈ (𝐴 ∩ ℝ) ↔ ((𝑥 + 1) ∈ 𝐴 ∧ (𝑥 + 1) ∈ ℝ))
13	12	simplbi2com 1444	. . . . . 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 2538	. . . 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 7944	. . . . 5 ⊢ ℝ ∈ V
18	17	inex2 4138	. . . 4 ⊢ (𝐴 ∩ ℝ) ∈ V
19		eleq2 2241	. . . . . . 7 ⊢ (𝑦 = (𝐴 ∩ ℝ) → (1 ∈ 𝑦 ↔ 1 ∈ (𝐴 ∩ ℝ)))
20		eleq2 2241	. . . . . . . 8 ⊢ (𝑦 = (𝐴 ∩ ℝ) → ((𝑥 + 1) ∈ 𝑦 ↔ (𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
21	20	raleqbi1dv 2680	. . . . . . 7 ⊢ (𝑦 = (𝐴 ∩ ℝ) → (∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ)))
22	19, 21	anbi12d 473	. . . . . 6 ⊢ (𝑦 = (𝐴 ∩ ℝ) → ((1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦) ↔ (1 ∈ (𝐴 ∩ ℝ) ∧ ∀𝑥 ∈ (𝐴 ∩ ℝ)(𝑥 + 1) ∈ (𝐴 ∩ ℝ))))
23	22	elabg 2883	. . . . 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 8920	. . . . . 6 ⊢ ℕ = ∩ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)}
25		intss1 3859	. . . . . 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 3201	. . . . 5 ⊢ ((𝐴 ∩ ℝ) ∈ {𝑦 ∣ (1 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 (𝑥 + 1) ∈ 𝑦)} → ℕ ⊆ (𝐴 ∩ ℝ))
27	23, 26	syl6bir 164	. . . 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 289	. 2 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ (𝐴 ∩ ℝ))
30	29, 5	sstrdi 3167	1 ⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  Vcvv 2737   ∩ cin 3128   ⊆ wss 3129  ∩ cint 3844  (class class class)co 5874  ℝcr 7809  1c1 7811   + caddc 7813  ℕcn 8918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4121  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142  df-int 3845  df-inn 8919

This theorem is referenced by:  nnssre  8922  nnind  8934