Theorem peano5 4441

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 4446. (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 4435	. . 3 ⊢ ω = ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)}
2		peano1 4437	. . . . . . . 8 ⊢ ∅ ∈ ω
3		elin 3198	. . . . . . . 8 ⊢ (∅ ∈ (ω ∩ 𝐴) ↔ (∅ ∈ ω ∧ ∅ ∈ 𝐴))
4	2, 3	mpbiran 889	. . . . . . 7 ⊢ (∅ ∈ (ω ∩ 𝐴) ↔ ∅ ∈ 𝐴)
5	4	biimpri 132	. . . . . 6 ⊢ (∅ ∈ 𝐴 → ∅ ∈ (ω ∩ 𝐴))
6		peano2 4438	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
7	6	adantr 271	. . . . . . . . . . 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		pm3.31 259	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ((𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ∈ 𝐴))
10	8, 9	jcad 302	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ((𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥 ∈ 𝐴)))
11	10	alimi 1396	. . . . . . . 8 ⊢ (∀𝑥(𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ∀𝑥((𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥 ∈ 𝐴)))
12		df-ral 2375	. . . . . . . 8 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
13		elin 3198	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ω ∩ 𝐴) ↔ (𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴))
14		elin 3198	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ (ω ∩ 𝐴) ↔ (suc 𝑥 ∈ ω ∧ suc 𝑥 ∈ 𝐴))
15	13, 14	imbi12i 238	. . . . . . . . 9 ⊢ ((𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)) ↔ ((𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥 ∈ 𝐴)))
16	15	albii 1411	. . . . . . . 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 200	. . . . . . 7 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
18		df-ral 2375	. . . . . . 7 ⊢ (∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴) ↔ ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
19	17, 18	sylibr 133	. . . . . 6 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴))
20	5, 19	anim12i 332	. . . . 5 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
21		omex 4436	. . . . . . 7 ⊢ ω ∈ V
22	21	inex1 3994	. . . . . 6 ⊢ (ω ∩ 𝐴) ∈ V
23		eleq2 2158	. . . . . . 7 ⊢ (𝑦 = (ω ∩ 𝐴) → (∅ ∈ 𝑦 ↔ ∅ ∈ (ω ∩ 𝐴)))
24		eleq2 2158	. . . . . . . 8 ⊢ (𝑦 = (ω ∩ 𝐴) → (suc 𝑥 ∈ 𝑦 ↔ suc 𝑥 ∈ (ω ∩ 𝐴)))
25	24	raleqbi1dv 2584	. . . . . . 7 ⊢ (𝑦 = (ω ∩ 𝐴) → (∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦 ↔ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
26	23, 25	anbi12d 458	. . . . . 6 ⊢ (𝑦 = (ω ∩ 𝐴) → ((∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) ↔ (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴))))
27	22, 26	elab 2774	. . . . 5 ⊢ ((ω ∩ 𝐴) ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} ↔ (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
28	20, 27	sylibr 133	. . . 4 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (ω ∩ 𝐴) ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)})
29		intss1 3725	. . . 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 3085	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ (ω ∩ 𝐴))
32		ssid 3059	. . . 4 ⊢ ω ⊆ ω
33	32	biantrur 298	. . 3 ⊢ (ω ⊆ 𝐴 ↔ (ω ⊆ ω ∧ ω ⊆ 𝐴))
34		ssin 3237	. . 3 ⊢ ((ω ⊆ ω ∧ ω ⊆ 𝐴) ↔ ω ⊆ (ω ∩ 𝐴))
35	33, 34	bitri 183	. 2 ⊢ (ω ⊆ 𝐴 ↔ ω ⊆ (ω ∩ 𝐴))
36	31, 35	sylibr 133	1 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1294   = wceq 1296   ∈ wcel 1445  {cab 2081  ∀wral 2370   ∩ cin 3012   ⊆ wss 3013  ∅c0 3302  ∩ cint 3710  suc csuc 4216  ωcom 4433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-int 3711  df-suc 4222  df-iom 4434

This theorem is referenced by:  find  4442  finds  4443  finds2  4444  indpi  6998