Theorem peano5 4594

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 4599. (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 4588	. . 3 ⊢ ω = ∩ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥 ∈ 𝑦 suc 𝑥 ∈ 𝑦)}
2		peano1 4590	. . . . . . . 8 ⊢ ∅ ∈ ω
3		elin 3318	. . . . . . . 8 ⊢ (∅ ∈ (ω ∩ 𝐴) ↔ (∅ ∈ ω ∧ ∅ ∈ 𝐴))
4	2, 3	mpbiran 940	. . . . . . 7 ⊢ (∅ ∈ (ω ∩ 𝐴) ↔ ∅ ∈ 𝐴)
5	4	biimpri 133	. . . . . 6 ⊢ (∅ ∈ 𝐴 → ∅ ∈ (ω ∩ 𝐴))
6		peano2 4591	. . . . . . . . . . . 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 1455	. . . . . . . 8 ⊢ (∀𝑥(𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ∀𝑥((𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥 ∈ 𝐴)))
12		df-ral 2460	. . . . . . . 8 ⊢ (∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ ω → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
13		elin 3318	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ω ∩ 𝐴) ↔ (𝑥 ∈ ω ∧ 𝑥 ∈ 𝐴))
14		elin 3318	. . . . . . . . . 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 1470	. . . . . . . 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 2460	. . . . . . 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 4589	. . . . . . 7 ⊢ ω ∈ V
22	21	inex1 4134	. . . . . 6 ⊢ (ω ∩ 𝐴) ∈ V
23		eleq2 2241	. . . . . . 7 ⊢ (𝑦 = (ω ∩ 𝐴) → (∅ ∈ 𝑦 ↔ ∅ ∈ (ω ∩ 𝐴)))
24		eleq2 2241	. . . . . . . 8 ⊢ (𝑦 = (ω ∩ 𝐴) → (suc 𝑥 ∈ 𝑦 ↔ suc 𝑥 ∈ (ω ∩ 𝐴)))
25	24	raleqbi1dv 2680	. . . . . . 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 2881	. . . . 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 3857	. . . 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 3201	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ (ω ∩ 𝐴))
32		ssid 3175	. . . 4 ⊢ ω ⊆ ω
33	32	biantrur 303	. . 3 ⊢ (ω ⊆ 𝐴 ↔ (ω ⊆ ω ∧ ω ⊆ 𝐴))
34		ssin 3357	. . 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 1351   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455   ∩ cin 3128   ⊆ wss 3129  ∅c0 3422  ∩ cint 3842  suc csuc 4362  ωcom 4586

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 614  ax-in2 615  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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-3an 980  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-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808  df-int 3843  df-suc 4368  df-iom 4587

This theorem is referenced by:  find  4595  finds  4596  finds2  4597  indpi  7332