Theorem indpi 7150

Description: Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.)

Hypotheses

Ref	Expression
indpi.1	⊢ (𝑥 = 1o → (𝜑 ↔ 𝜓))
indpi.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
indpi.3	⊢ (𝑥 = (𝑦 +N 1o) → (𝜑 ↔ 𝜃))
indpi.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
indpi.5	⊢ 𝜓
indpi.6	⊢ (𝑦 ∈ N → (𝜒 → 𝜃))

Assertion

Ref	Expression
indpi	⊢ (𝐴 ∈ N → 𝜏)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem indpi

Step	Hyp	Ref	Expression
1		indpi.4	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
2		elni 7116	. . 3 ⊢ (𝑥 ∈ N ↔ (𝑥 ∈ ω ∧ 𝑥 ≠ ∅))
3		eqid 2139	. . . . . . . . . 10 ⊢ ∅ = ∅
4	3	orci 720	. . . . . . . . 9 ⊢ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)
5		nfv 1508	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∅ = ∅
6		nfsbc1v 2927	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑
7	5, 6	nfor 1553	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∅ = ∅ ∨ [∅ / 𝑥]𝜑)
8		0ex 4055	. . . . . . . . . 10 ⊢ ∅ ∈ V
9		eqeq1 2146	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
10		sbceq1a 2918	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑))
11	9, 10	orbi12d 782	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)))
12	7, 8, 11	elabf 2827	. . . . . . . . 9 ⊢ (∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑))
13	4, 12	mpbir 145	. . . . . . . 8 ⊢ ∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
14		suceq 4324	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
15		df-1o 6313	. . . . . . . . . . . . . 14 ⊢ 1o = suc ∅
16	14, 15	syl6eqr 2190	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → suc 𝑦 = 1o)
17		indpi.5	. . . . . . . . . . . . . . 15 ⊢ 𝜓
18	17	olci 721	. . . . . . . . . . . . . 14 ⊢ (1o = ∅ ∨ 𝜓)
19		1oex 6321	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
20		eqeq1 2146	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1o → (𝑥 = ∅ ↔ 1o = ∅))
21		indpi.1	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1o → (𝜑 ↔ 𝜓))
22	20, 21	orbi12d 782	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1o → ((𝑥 = ∅ ∨ 𝜑) ↔ (1o = ∅ ∨ 𝜓)))
23	19, 22	elab 2828	. . . . . . . . . . . . . 14 ⊢ (1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (1o = ∅ ∨ 𝜓))
24	18, 23	mpbir 145	. . . . . . . . . . . . 13 ⊢ 1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
25	16, 24	eqeltrdi 2230	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
26	25	a1d 22	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
27	26	a1i 9	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})))
28		indpi.6	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ N → (𝜒 → 𝜃))
29		elni 7116	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
30	29	simprbi 273	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ N → 𝑦 ≠ ∅)
31	30	neneqd 2329	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ N → ¬ 𝑦 = ∅)
32		biorf 733	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑦 = ∅ → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
33	31, 32	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ N → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
34		vex 2689	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
35		eqeq1 2146	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
36		indpi.2	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
37	35, 36	orbi12d 782	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝜑) ↔ (𝑦 = ∅ ∨ 𝜒)))
38	34, 37	elab 2828	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑦 = ∅ ∨ 𝜒))
39	33, 38	syl6bbr 197	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ N → (𝜒 ↔ 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
40		1pi 7123	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ∈ N
41		addclpi 7135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ N ∧ 1o ∈ N) → (𝑦 +N 1o) ∈ N)
42	40, 41	mpan2 421	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ N → (𝑦 +N 1o) ∈ N)
43		elni 7116	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 +N 1o) ∈ N ↔ ((𝑦 +N 1o) ∈ ω ∧ (𝑦 +N 1o) ≠ ∅))
44	42, 43	sylib 121	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ N → ((𝑦 +N 1o) ∈ ω ∧ (𝑦 +N 1o) ≠ ∅))
45	44	simprd 113	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ N → (𝑦 +N 1o) ≠ ∅)
46	45	neneqd 2329	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ N → ¬ (𝑦 +N 1o) = ∅)
47		biorf 733	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑦 +N 1o) = ∅ → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
48	46, 47	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ N → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
49		eqeq1 2146	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 +N 1o) → (𝑥 = ∅ ↔ (𝑦 +N 1o) = ∅))
50		indpi.3	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 +N 1o) → (𝜑 ↔ 𝜃))
51	49, 50	orbi12d 782	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 +N 1o) → ((𝑥 = ∅ ∨ 𝜑) ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
52	51	elabg 2830	. . . . . . . . . . . . . 14 ⊢ ((𝑦 +N 1o) ∈ N → ((𝑦 +N 1o) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
53	42, 52	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ N → ((𝑦 +N 1o) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
54		addpiord 7124	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ N ∧ 1o ∈ N) → (𝑦 +N 1o) = (𝑦 +o 1o))
55	40, 54	mpan2 421	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ N → (𝑦 +N 1o) = (𝑦 +o 1o))
56		pion 7118	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ N → 𝑦 ∈ On)
57		oa1suc 6363	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → (𝑦 +o 1o) = suc 𝑦)
58	56, 57	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ N → (𝑦 +o 1o) = suc 𝑦)
59	55, 58	eqtrd 2172	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ N → (𝑦 +N 1o) = suc 𝑦)
60	59	eleq1d 2208	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ N → ((𝑦 +N 1o) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
61	48, 53, 60	3bitr2d 215	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ N → (𝜃 ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
62	28, 39, 61	3imtr3d 201	. . . . . . . . . . 11 ⊢ (𝑦 ∈ N → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
63	62	a1i 9	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝑦 ∈ N → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})))
64		nndceq0 4531	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → DECID 𝑦 = ∅)
65		df-dc 820	. . . . . . . . . . . 12 ⊢ (DECID 𝑦 = ∅ ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
66	64, 65	sylib 121	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
67		idd 21	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → 𝑦 = ∅))
68	67	necon3bd 2351	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 ≠ ∅))
69	68	anc2li 327	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → (¬ 𝑦 = ∅ → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
70	69, 29	syl6ibr 161	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 ∈ N))
71	70	orim2d 777	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (𝑦 = ∅ ∨ 𝑦 ∈ N)))
72	66, 71	mpd 13	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦 ∈ N))
73	27, 63, 72	mpjaod 707	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
74	73	rgen 2485	. . . . . . . 8 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
75		peano5 4512	. . . . . . . 8 ⊢ ((∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) → ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
76	13, 74, 75	mp2an 422	. . . . . . 7 ⊢ ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
77	76	sseli 3093	. . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
78		abid 2127	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑥 = ∅ ∨ 𝜑))
79	77, 78	sylib 121	. . . . 5 ⊢ (𝑥 ∈ ω → (𝑥 = ∅ ∨ 𝜑))
80	79	adantr 274	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝑥 = ∅ ∨ 𝜑))
81		df-ne 2309	. . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
82		biorf 733	. . . . . 6 ⊢ (¬ 𝑥 = ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
83	81, 82	sylbi 120	. . . . 5 ⊢ (𝑥 ≠ ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
84	83	adantl 275	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
85	80, 84	mpbird 166	. . 3 ⊢ ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → 𝜑)
86	2, 85	sylbi 120	. 2 ⊢ (𝑥 ∈ N → 𝜑)
87	1, 86	vtoclga 2752	1 ⊢ (𝐴 ∈ N → 𝜏)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  DECID wdc 819   = wceq 1331   ∈ wcel 1480  {cab 2125   ≠ wne 2308  ∀wral 2416  [wsbc 2909   ⊆ wss 3071  ∅c0 3363  Oncon0 4285  suc csuc 4287  ωcom 4504  (class class class)co 5774  1_oc1o 6306   +_o coa 6310  Ncnpi 7080   +_N cpli 7081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-ni 7112  df-pli 7113

This theorem is referenced by:  pitonn  7656