Theorem peano5 4630

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 4635. (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 4624	. . 3 ⊢ ω = ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)}
2		peano1 4626	. . . . . . . 8 ⊢ ∅ ∈ ω
3		elin 3342	. . . . . . . 8 ⊢ (∅ ∈ (ω ∩ 𝐴) ↔ (∅ ∈ ω ∧ ∅ ∈ 𝐴))
4	2, 3	mpbiran 942	. . . . . . 7 ⊢ (∅ ∈ (ω ∩ 𝐴) ↔ ∅ ∈ 𝐴)
5	4	biimpri 133	. . . . . 6 ⊢ (∅ ∈ 𝐴 → ∅ ∈ (ω ∩ 𝐴))
6		peano2 4627	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
7	6	adantr 276	. . . . . . . . . . 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		pm3.31 262	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ((𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ∈ 𝐴))
10	8, 9	jcad 307	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ((𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥 ∈ 𝐴)))
11	10	alimi 1466	. . . . . . . 8 ⊢ (∀𝑥(𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ∀𝑥((𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥 ∈ 𝐴)))
12		df-ral 2477	. . . . . . . 8 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
13		elin 3342	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ω ∩ 𝐴) ↔ (𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴))
14		elin 3342	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ (ω ∩ 𝐴) ↔ (suc 𝑥 ∈ ω ∧ suc 𝑥 ∈ 𝐴))
15	13, 14	imbi12i 239	. . . . . . . . 9 ⊢ ((𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)) ↔ ((𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥 ∈ 𝐴)))
16	15	albii 1481	. . . . . . . 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 201	. . . . . . 7 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
18		df-ral 2477	. . . . . . 7 ⊢ (∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴) ↔ ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
19	17, 18	sylibr 134	. . . . . 6 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴))
20	5, 19	anim12i 338	. . . . 5 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
21		omex 4625	. . . . . . 7 ⊢ ω ∈ V
22	21	inex1 4163	. . . . . 6 ⊢ (ω ∩ 𝐴) ∈ V
23		eleq2 2257	. . . . . . 7 ⊢ (𝑦 = (ω ∩ 𝐴) → (∅ ∈ 𝑦 ↔ ∅ ∈ (ω ∩ 𝐴)))
24		eleq2 2257	. . . . . . . 8 ⊢ (𝑦 = (ω ∩ 𝐴) → (suc 𝑥 ∈ 𝑦 ↔ suc 𝑥 ∈ (ω ∩ 𝐴)))
25	24	raleqbi1dv 2702	. . . . . . 7 ⊢ (𝑦 = (ω ∩ 𝐴) → (∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦 ↔ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
26	23, 25	anbi12d 473	. . . . . 6 ⊢ (𝑦 = (ω ∩ 𝐴) → ((∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦) ↔ (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴))))
27	22, 26	elab 2904	. . . . 5 ⊢ ((ω ∩ 𝐴) ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)} ↔ (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
28	20, 27	sylibr 134	. . . 4 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (ω ∩ 𝐴) ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)})
29		intss1 3885	. . . 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	eqsstrid 3225	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ (ω ∩ 𝐴))
32		ssid 3199	. . . 4 ⊢ ω ⊆ ω
33	32	biantrur 303	. . 3 ⊢ (ω ⊆ 𝐴 ↔ (ω ⊆ ω ∧ ω ⊆ 𝐴))
34		ssin 3381	. . 3 ⊢ ((ω ⊆ ω ∧ ω ⊆ 𝐴) ↔ ω ⊆ (ω ∩ 𝐴))
35	33, 34	bitri 184	. 2 ⊢ (ω ⊆ 𝐴 ↔ ω ⊆ (ω ∩ 𝐴))
36	31, 35	sylibr 134	1 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472   ∩ cin 3152   ⊆ wss 3153  ∅c0 3446  ∩ cint 3870  suc csuc 4396  ωcom 4622

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-suc 4402  df-iom 4623

This theorem is referenced by:  find  4631  finds  4632  finds2  4633  indpi  7402