Theorem indpi 7409

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 7375	. . 3 ⊢ (𝑥 ∈ N ↔ (𝑥 ∈ ω ∧ 𝑥 ≠ ∅))
3		eqid 2196	. . . . . . . . . 10 ⊢ ∅ = ∅
4	3	orci 732	. . . . . . . . 9 ⊢ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)
5		nfv 1542	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∅ = ∅
6		nfsbc1v 3008	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑
7	5, 6	nfor 1588	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∅ = ∅ ∨ [∅ / 𝑥]𝜑)
8		0ex 4160	. . . . . . . . . 10 ⊢ ∅ ∈ V
9		eqeq1 2203	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
10		sbceq1a 2999	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑))
11	9, 10	orbi12d 794	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑)))
12	7, 8, 11	elabf 2907	. . . . . . . . 9 ⊢ (∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ = ∅ ∨ [∅ / 𝑥]𝜑))
13	4, 12	mpbir 146	. . . . . . . 8 ⊢ ∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
14		suceq 4437	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
15		df-1o 6474	. . . . . . . . . . . . . 14 ⊢ 1o = suc ∅
16	14, 15	eqtr4di 2247	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → suc 𝑦 = 1o)
17		indpi.5	. . . . . . . . . . . . . . 15 ⊢ 𝜓
18	17	olci 733	. . . . . . . . . . . . . 14 ⊢ (1o = ∅ ∨ 𝜓)
19		1oex 6482	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
20		eqeq1 2203	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1o → (𝑥 = ∅ ↔ 1o = ∅))
21		indpi.1	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1o → (𝜑 ↔ 𝜓))
22	20, 21	orbi12d 794	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1o → ((𝑥 = ∅ ∨ 𝜑) ↔ (1o = ∅ ∨ 𝜓)))
23	19, 22	elab 2908	. . . . . . . . . . . . . 14 ⊢ (1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (1o = ∅ ∨ 𝜓))
24	18, 23	mpbir 146	. . . . . . . . . . . . 13 ⊢ 1o ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
25	16, 24	eqeltrdi 2287	. . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ N ↔ (𝑦 ∈ ω ∧ 𝑦 ≠ ∅))
30	29	simprbi 275	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ N → 𝑦 ≠ ∅)
31	30	neneqd 2388	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ N → ¬ 𝑦 = ∅)
32		biorf 745	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑦 = ∅ → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
33	31, 32	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ N → (𝜒 ↔ (𝑦 = ∅ ∨ 𝜒)))
34		vex 2766	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
35		eqeq1 2203	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
36		indpi.2	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
37	35, 36	orbi12d 794	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ 𝜑) ↔ (𝑦 = ∅ ∨ 𝜒)))
38	34, 37	elab 2908	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑦 = ∅ ∨ 𝜒))
39	33, 38	bitr4di 198	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ N → (𝜒 ↔ 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
40		1pi 7382	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ∈ N
41		addclpi 7394	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ N ∧ 1o ∈ N) → (𝑦 +N 1o) ∈ N)
42	40, 41	mpan2 425	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ N → (𝑦 +N 1o) ∈ N)
43		elni 7375	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 +N 1o) ∈ N ↔ ((𝑦 +N 1o) ∈ ω ∧ (𝑦 +N 1o) ≠ ∅))
44	42, 43	sylib 122	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ N → ((𝑦 +N 1o) ∈ ω ∧ (𝑦 +N 1o) ≠ ∅))
45	44	simprd 114	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ N → (𝑦 +N 1o) ≠ ∅)
46	45	neneqd 2388	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ N → ¬ (𝑦 +N 1o) = ∅)
47		biorf 745	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑦 +N 1o) = ∅ → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
48	46, 47	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ N → (𝜃 ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
49		eqeq1 2203	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 +N 1o) → (𝑥 = ∅ ↔ (𝑦 +N 1o) = ∅))
50		indpi.3	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 +N 1o) → (𝜑 ↔ 𝜃))
51	49, 50	orbi12d 794	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 +N 1o) → ((𝑥 = ∅ ∨ 𝜑) ↔ ((𝑦 +N 1o) = ∅ ∨ 𝜃)))
52	51	elabg 2910	. . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ N ∧ 1o ∈ N) → (𝑦 +N 1o) = (𝑦 +o 1o))
55	40, 54	mpan2 425	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ N → (𝑦 +N 1o) = (𝑦 +o 1o))
56		pion 7377	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ N → 𝑦 ∈ On)
57		oa1suc 6525	. . . . . . . . . . . . . . . 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 2229	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ N → (𝑦 +N 1o) = suc 𝑦)
60	59	eleq1d 2265	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ N → ((𝑦 +N 1o) ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
61	48, 53, 60	3bitr2d 216	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ N → (𝜃 ↔ suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
62	28, 39, 61	3imtr3d 202	. . . . . . . . . . 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 4654	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → DECID 𝑦 = ∅)
65		df-dc 836	. . . . . . . . . . . 12 ⊢ (DECID 𝑦 = ∅ ↔ (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
66	64, 65	sylib 122	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ ¬ 𝑦 = ∅))
67		idd 21	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → 𝑦 = ∅))
68	67	necon3bd 2410	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 ≠ ∅))
69	68	anc2li 329	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → (¬ 𝑦 = ∅ → (𝑦 ∈ ω ∧ 𝑦 ≠ ∅)))
70	69, 29	imbitrrdi 162	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 ∈ N))
71	70	orim2d 789	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ¬ 𝑦 = ∅) → (𝑦 = ∅ ∨ 𝑦 ∈ N)))
72	66, 71	mpd 13	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦 ∈ N))
73	27, 63, 72	mpjaod 719	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}))
74	73	rgen 2550	. . . . . . . 8 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
75		peano5 4634	. . . . . . . 8 ⊢ ((∅ ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})) → ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
76	13, 74, 75	mp2an 426	. . . . . . 7 ⊢ ω ⊆ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)}
77	76	sseli 3179	. . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)})
78		abid 2184	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (𝑥 = ∅ ∨ 𝜑))
79	77, 78	sylib 122	. . . . 5 ⊢ (𝑥 ∈ ω → (𝑥 = ∅ ∨ 𝜑))
80	79	adantr 276	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝑥 = ∅ ∨ 𝜑))
81		df-ne 2368	. . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
82		biorf 745	. . . . . 6 ⊢ (¬ 𝑥 = ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
83	81, 82	sylbi 121	. . . . 5 ⊢ (𝑥 ≠ ∅ → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
84	83	adantl 277	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → (𝜑 ↔ (𝑥 = ∅ ∨ 𝜑)))
85	80, 84	mpbird 167	. . 3 ⊢ ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → 𝜑)
86	2, 85	sylbi 121	. 2 ⊢ (𝑥 ∈ N → 𝜑)
87	1, 86	vtoclga 2830	1 ⊢ (𝐴 ∈ N → 𝜏)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2167  {cab 2182   ≠ wne 2367  ∀wral 2475  [wsbc 2989   ⊆ wss 3157  ∅c0 3450  Oncon0 4398  suc csuc 4400  ωcom 4626  (class class class)co 5922  1_oc1o 6467   +_o coa 6471  Ncnpi 7339   +_N cpli 7340

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-ni 7371  df-pli 7372

This theorem is referenced by:  pitonn  7915