Theorem indpi 7283

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 7249	. . 3 ⊢ (𝑥 ∈ N ↔ (𝑥 ∈ ω ∧ 𝑥 ≠ ∅))
3		eqid 2165	. . . . . . . . . 10 ⊢ ∅ = ∅
4	3	orci 721	. . . . . . . . 9 ⊢ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)
5		nfv 1516	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∅ = ∅
6		nfsbc1v 2969	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑
7	5, 6	nfor 1562	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∅ = ∅ ∨ [∅ / 𝑥]𝜑)
8		0ex 4109	. . . . . . . . . 10 ⊢ ∅ ∈ V
9		eqeq1 2172	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
10		sbceq1a 2960	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑))
11	9, 10	orbi12d 783	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)))
12	7, 8, 11	elabf 2869	. . . . . . . . 9 ⊢ (∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑))
13	4, 12	mpbir 145	. . . . . . . 8 ⊢ ∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
14		suceq 4380	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
15		df-1o 6384	. . . . . . . . . . . . . 14 ⊢ 1o = suc ∅
16	14, 15	eqtr4di 2217	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → suc 𝑦 = 1o)
17		indpi.5	. . . . . . . . . . . . . . 15 ⊢ 𝜓
18	17	olci 722	. . . . . . . . . . . . . 14 ⊢ (1o = ∅ ∨ 𝜓)
19		1oex 6392	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
20		eqeq1 2172	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1o → (𝑥 = ∅ ↔ 1o = ∅))
21		indpi.1	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1o → (𝜑 ↔ 𝜓))
22	20, 21	orbi12d 783	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1o → ((𝑥 = ∅ ∨ 𝜑) ↔ (1o = ∅ ∨ 𝜓)))
23	19, 22	elab 2870	. . . . . . . . . . . . . 14 ⊢ (1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (1o = ∅ ∨ 𝜓))
24	18, 23	mpbir 145	. . . . . . . . . . . . 13 ⊢ 1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
25	16, 24	eqeltrdi 2257	. . . . . . . . . . . 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 7249	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
30	29	simprbi 273	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ N → 𝑦 ≠ ∅)
31	30	neneqd 2357	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ N → ¬ 𝑦 = ∅)
32		biorf 734	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑦 = ∅ → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
33	31, 32	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ N → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
34		vex 2729	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
35		eqeq1 2172	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
36		indpi.2	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
37	35, 36	orbi12d 783	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝜑) ↔ (𝑦 = ∅ ∨ 𝜒)))
38	34, 37	elab 2870	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑦 = ∅ ∨ 𝜒))
39	33, 38	bitr4di 197	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ N → (𝜒 ↔ 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
40		1pi 7256	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ∈ N
41		addclpi 7268	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ N ∧ 1o ∈ N) → (𝑦 +N 1o) ∈ N)
42	40, 41	mpan2 422	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ N → (𝑦 +N 1o) ∈ N)
43		elni 7249	. . . . . . . . . . . . . . . . 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 2357	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ N → ¬ (𝑦 +N 1o) = ∅)
47		biorf 734	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑦 +N 1o) = ∅ → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
48	46, 47	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ N → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
49		eqeq1 2172	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 +N 1o) → (𝑥 = ∅ ↔ (𝑦 +N 1o) = ∅))
50		indpi.3	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 +N 1o) → (𝜑 ↔ 𝜃))
51	49, 50	orbi12d 783	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 +N 1o) → ((𝑥 = ∅ ∨ 𝜑) ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
52	51	elabg 2872	. . . . . . . . . . . . . 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 7257	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ N ∧ 1o ∈ N) → (𝑦 +N 1o) = (𝑦 +o 1o))
55	40, 54	mpan2 422	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ N → (𝑦 +N 1o) = (𝑦 +o 1o))
56		pion 7251	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ N → 𝑦 ∈ On)
57		oa1suc 6435	. . . . . . . . . . . . . . . 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 2198	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ N → (𝑦 +N 1o) = suc 𝑦)
60	59	eleq1d 2235	. . . . . . . . . . . . 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 4595	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → DECID 𝑦 = ∅)
65		df-dc 825	. . . . . . . . . . . 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 2379	. . . . . . . . . . . . . 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 778	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (𝑦 = ∅ ∨ 𝑦 ∈ N)))
72	66, 71	mpd 13	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦 ∈ N))
73	27, 63, 72	mpjaod 708	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
74	73	rgen 2519	. . . . . . . 8 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
75		peano5 4575	. . . . . . . 8 ⊢ ((∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) → ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
76	13, 74, 75	mp2an 423	. . . . . . 7 ⊢ ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
77	76	sseli 3138	. . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
78		abid 2153	. . . . . 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 2337	. . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
82		biorf 734	. . . . . 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 2792	1 ⊢ (𝐴 ∈ N → 𝜏)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 824   = wceq 1343   ∈ wcel 2136  {cab 2151   ≠ wne 2336  ∀wral 2444  [wsbc 2951   ⊆ wss 3116  ∅c0 3409  Oncon0 4341  suc csuc 4343  ωcom 4567  (class class class)co 5842  1_oc1o 6377   +_o coa 6381  Ncnpi 7213   +_N cpli 7214

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-ni 7245  df-pli 7246

This theorem is referenced by:  pitonn  7789