Theorem peano5 4264

Description: The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as theorem findes 4269. (Contributed by NM, 18-Feb-2004.)

Assertion

Ref	Expression
peano5	⊢ ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → 𝜔 ⊆ A)

Distinct variable group:   x,A

Proof of Theorem peano5

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfom3 4258	. . 3 ⊢ 𝜔 = ∩ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)}
2		peano1 4260	. . . . . . . 8 ⊢ ∅ ∈ 𝜔
3		elin 3120	. . . . . . . 8 ⊢ (∅ ∈ (𝜔 ∩ A) ↔ (∅ ∈ 𝜔 ∧ ∅ ∈ A))
4	2, 3	mpbiran 846	. . . . . . 7 ⊢ (∅ ∈ (𝜔 ∩ A) ↔ ∅ ∈ A)
5	4	biimpri 124	. . . . . 6 ⊢ (∅ ∈ A → ∅ ∈ (𝜔 ∩ A))
6		peano2 4261	. . . . . . . . . . . 12 ⊢ (x ∈ 𝜔 → suc x ∈ 𝜔)
7	6	adantr 261	. . . . . . . . . . 11 ⊢ ((x ∈ 𝜔 ∧ x ∈ A) → suc x ∈ 𝜔)
8	7	a1i 9	. . . . . . . . . 10 ⊢ ((x ∈ 𝜔 → (x ∈ A → suc x ∈ A)) → ((x ∈ 𝜔 ∧ x ∈ A) → suc x ∈ 𝜔))
9		pm3.31 249	. . . . . . . . . 10 ⊢ ((x ∈ 𝜔 → (x ∈ A → suc x ∈ A)) → ((x ∈ 𝜔 ∧ x ∈ A) → suc x ∈ A))
10	8, 9	jcad 291	. . . . . . . . 9 ⊢ ((x ∈ 𝜔 → (x ∈ A → suc x ∈ A)) → ((x ∈ 𝜔 ∧ x ∈ A) → (suc x ∈ 𝜔 ∧ suc x ∈ A)))
11	10	alimi 1341	. . . . . . . 8 ⊢ (∀x(x ∈ 𝜔 → (x ∈ A → suc x ∈ A)) → ∀x((x ∈ 𝜔 ∧ x ∈ A) → (suc x ∈ 𝜔 ∧ suc x ∈ A)))
12		df-ral 2305	. . . . . . . 8 ⊢ (∀x ∈ 𝜔 (x ∈ A → suc x ∈ A) ↔ ∀x(x ∈ 𝜔 → (x ∈ A → suc x ∈ A)))
13		elin 3120	. . . . . . . . . 10 ⊢ (x ∈ (𝜔 ∩ A) ↔ (x ∈ 𝜔 ∧ x ∈ A))
14		elin 3120	. . . . . . . . . 10 ⊢ (suc x ∈ (𝜔 ∩ A) ↔ (suc x ∈ 𝜔 ∧ suc x ∈ A))
15	13, 14	imbi12i 228	. . . . . . . . 9 ⊢ ((x ∈ (𝜔 ∩ A) → suc x ∈ (𝜔 ∩ A)) ↔ ((x ∈ 𝜔 ∧ x ∈ A) → (suc x ∈ 𝜔 ∧ suc x ∈ A)))
16	15	albii 1356	. . . . . . . 8 ⊢ (∀x(x ∈ (𝜔 ∩ A) → suc x ∈ (𝜔 ∩ A)) ↔ ∀x((x ∈ 𝜔 ∧ x ∈ A) → (suc x ∈ 𝜔 ∧ suc x ∈ A)))
17	11, 12, 16	3imtr4i 190	. . . . . . 7 ⊢ (∀x ∈ 𝜔 (x ∈ A → suc x ∈ A) → ∀x(x ∈ (𝜔 ∩ A) → suc x ∈ (𝜔 ∩ A)))
18		df-ral 2305	. . . . . . 7 ⊢ (∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A) ↔ ∀x(x ∈ (𝜔 ∩ A) → suc x ∈ (𝜔 ∩ A)))
19	17, 18	sylibr 137	. . . . . 6 ⊢ (∀x ∈ 𝜔 (x ∈ A → suc x ∈ A) → ∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A))
20	5, 19	anim12i 321	. . . . 5 ⊢ ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → (∅ ∈ (𝜔 ∩ A) ∧ ∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A)))
21		omex 4259	. . . . . . 7 ⊢ 𝜔 ∈ V
22	21	inex1 3882	. . . . . 6 ⊢ (𝜔 ∩ A) ∈ V
23		eleq2 2098	. . . . . . 7 ⊢ (y = (𝜔 ∩ A) → (∅ ∈ y ↔ ∅ ∈ (𝜔 ∩ A)))
24		eleq2 2098	. . . . . . . 8 ⊢ (y = (𝜔 ∩ A) → (suc x ∈ y ↔ suc x ∈ (𝜔 ∩ A)))
25	24	raleqbi1dv 2507	. . . . . . 7 ⊢ (y = (𝜔 ∩ A) → (∀x ∈ y suc x ∈ y ↔ ∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A)))
26	23, 25	anbi12d 442	. . . . . 6 ⊢ (y = (𝜔 ∩ A) → ((∅ ∈ y ∧ ∀x ∈ y suc x ∈ y) ↔ (∅ ∈ (𝜔 ∩ A) ∧ ∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A))))
27	22, 26	elab 2681	. . . . 5 ⊢ ((𝜔 ∩ A) ∈ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)} ↔ (∅ ∈ (𝜔 ∩ A) ∧ ∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A)))
28	20, 27	sylibr 137	. . . 4 ⊢ ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → (𝜔 ∩ A) ∈ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)})
29		intss1 3621	. . . 4 ⊢ ((𝜔 ∩ A) ∈ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)} → ∩ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)} ⊆ (𝜔 ∩ A))
30	28, 29	syl 14	. . 3 ⊢ ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → ∩ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)} ⊆ (𝜔 ∩ A))
31	1, 30	syl5eqss 2983	. 2 ⊢ ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → 𝜔 ⊆ (𝜔 ∩ A))
32		ssid 2958	. . . 4 ⊢ 𝜔 ⊆ 𝜔
33	32	biantrur 287	. . 3 ⊢ (𝜔 ⊆ A ↔ (𝜔 ⊆ 𝜔 ∧ 𝜔 ⊆ A))
34		ssin 3153	. . 3 ⊢ ((𝜔 ⊆ 𝜔 ∧ 𝜔 ⊆ A) ↔ 𝜔 ⊆ (𝜔 ∩ A))
35	33, 34	bitri 173	. 2 ⊢ (𝜔 ⊆ A ↔ 𝜔 ⊆ (𝜔 ∩ A))
36	31, 35	sylibr 137	1 ⊢ ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → 𝜔 ⊆ A)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300   ∩ cin 2910   ⊆ wss 2911  ∅c0 3218  ∩ cint 3606  suc csuc 4068  𝜔com 4256

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-in1 544  ax-in2 545  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-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  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-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257

This theorem is referenced by:  find  4265  finds  4266  finds2  4267  indpi  6326