Theorem omsinds 4658

Description: Strong (or "total") induction principle over ω. (Contributed by Scott Fenton, 17-Jul-2015.)

Hypotheses

Ref	Expression
omsinds.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
omsinds.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
omsinds.3	⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))

Assertion

Ref	Expression
omsinds	⊢ (𝐴 ∈ ω → 𝜒)

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

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

Proof of Theorem omsinds

Dummy variables 𝑤 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omsinds.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
2		suceq 4437	. . . 4 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅)
3	2	raleqdv 2699	. . 3 ⊢ (𝑤 = ∅ → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc ∅𝜑))
4		suceq 4437	. . . 4 ⊢ (𝑤 = 𝑘 → suc 𝑤 = suc 𝑘)
5	4	raleqdv 2699	. . 3 ⊢ (𝑤 = 𝑘 → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc 𝑘𝜑))
6		suceq 4437	. . . 4 ⊢ (𝑤 = suc 𝑘 → suc 𝑤 = suc suc 𝑘)
7	6	raleqdv 2699	. . 3 ⊢ (𝑤 = suc 𝑘 → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc suc 𝑘𝜑))
8		suceq 4437	. . . 4 ⊢ (𝑤 = 𝐴 → suc 𝑤 = suc 𝐴)
9	8	raleqdv 2699	. . 3 ⊢ (𝑤 = 𝐴 → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc 𝐴𝜑))
10		ral0 3552	. . . . . 6 ⊢ ∀𝑦 ∈ ∅ 𝜓
11		omsinds.3	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))
12	11	rgen 2550	. . . . . . 7 ⊢ ∀𝑥 ∈ ω (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)
13		peano1 4630	. . . . . . . 8 ⊢ ∅ ∈ ω
14	10	nfth 1478	. . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑦 ∈ ∅ 𝜓
15		nfsbc1v 3008	. . . . . . . . . 10 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑
16	14, 15	nfim 1586	. . . . . . . . 9 ⊢ Ⅎ𝑥(∀𝑦 ∈ ∅ 𝜓 → [∅ / 𝑥]𝜑)
17		raleq 2693	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓))
18		sbceq1a 2999	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑))
19	17, 18	imbi12d 234	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((∀𝑦 ∈ 𝑥 𝜓 → 𝜑) ↔ (∀𝑦 ∈ ∅ 𝜓 → [∅ / 𝑥]𝜑)))
20	16, 19	rspc 2862	. . . . . . . 8 ⊢ (∅ ∈ ω → (∀𝑥 ∈ ω (∀𝑦 ∈ 𝑥 𝜓 → 𝜑) → (∀𝑦 ∈ ∅ 𝜓 → [∅ / 𝑥]𝜑)))
21	13, 20	ax-mp 5	. . . . . . 7 ⊢ (∀𝑥 ∈ ω (∀𝑦 ∈ 𝑥 𝜓 → 𝜑) → (∀𝑦 ∈ ∅ 𝜓 → [∅ / 𝑥]𝜑))
22	12, 21	ax-mp 5	. . . . . 6 ⊢ (∀𝑦 ∈ ∅ 𝜓 → [∅ / 𝑥]𝜑)
23	10, 22	ax-mp 5	. . . . 5 ⊢ [∅ / 𝑥]𝜑
24		ralsns 3660	. . . . . 6 ⊢ (∅ ∈ ω → (∀𝑥 ∈ {∅}𝜑 ↔ [∅ / 𝑥]𝜑))
25	13, 24	ax-mp 5	. . . . 5 ⊢ (∀𝑥 ∈ {∅}𝜑 ↔ [∅ / 𝑥]𝜑)
26	23, 25	mpbir 146	. . . 4 ⊢ ∀𝑥 ∈ {∅}𝜑
27		suc0 4446	. . . . 5 ⊢ suc ∅ = {∅}
28	27	raleqi 2697	. . . 4 ⊢ (∀𝑥 ∈ suc ∅𝜑 ↔ ∀𝑥 ∈ {∅}𝜑)
29	26, 28	mpbir 146	. . 3 ⊢ ∀𝑥 ∈ suc ∅𝜑
30		simpr 110	. . . . . 6 ⊢ ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → ∀𝑥 ∈ suc 𝑘𝜑)
31		peano2 4631	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
32	31	adantr 276	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → suc 𝑘 ∈ ω)
33		omsinds.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
34	33	cbvralv 2729	. . . . . . . . 9 ⊢ (∀𝑥 ∈ suc 𝑘𝜑 ↔ ∀𝑦 ∈ suc 𝑘𝜓)
35	30, 34	sylib 122	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → ∀𝑦 ∈ suc 𝑘𝜓)
36		nfv 1542	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑦 ∈ suc 𝑘𝜓
37		nfsbc1v 3008	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[suc 𝑘 / 𝑥]𝜑
38	36, 37	nfim 1586	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∀𝑦 ∈ suc 𝑘𝜓 → [suc 𝑘 / 𝑥]𝜑)
39		raleq 2693	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑘 → (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑦 ∈ suc 𝑘𝜓))
40		sbceq1a 2999	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑘 → (𝜑 ↔ [suc 𝑘 / 𝑥]𝜑))
41	39, 40	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑘 → ((∀𝑦 ∈ 𝑥 𝜓 → 𝜑) ↔ (∀𝑦 ∈ suc 𝑘𝜓 → [suc 𝑘 / 𝑥]𝜑)))
42	38, 41	rspc 2862	. . . . . . . . 9 ⊢ (suc 𝑘 ∈ ω → (∀𝑥 ∈ ω (∀𝑦 ∈ 𝑥 𝜓 → 𝜑) → (∀𝑦 ∈ suc 𝑘𝜓 → [suc 𝑘 / 𝑥]𝜑)))
43	12, 42	mpi 15	. . . . . . . 8 ⊢ (suc 𝑘 ∈ ω → (∀𝑦 ∈ suc 𝑘𝜓 → [suc 𝑘 / 𝑥]𝜑))
44	32, 35, 43	sylc 62	. . . . . . 7 ⊢ ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → [suc 𝑘 / 𝑥]𝜑)
45		ralsns 3660	. . . . . . . 8 ⊢ (suc 𝑘 ∈ ω → (∀𝑥 ∈ {suc 𝑘}𝜑 ↔ [suc 𝑘 / 𝑥]𝜑))
46	32, 45	syl 14	. . . . . . 7 ⊢ ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → (∀𝑥 ∈ {suc 𝑘}𝜑 ↔ [suc 𝑘 / 𝑥]𝜑))
47	44, 46	mpbird 167	. . . . . 6 ⊢ ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → ∀𝑥 ∈ {suc 𝑘}𝜑)
48		ralun 3345	. . . . . 6 ⊢ ((∀𝑥 ∈ suc 𝑘𝜑 ∧ ∀𝑥 ∈ {suc 𝑘}𝜑) → ∀𝑥 ∈ (suc 𝑘 ∪ {suc 𝑘})𝜑)
49	30, 47, 48	syl2anc 411	. . . . 5 ⊢ ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → ∀𝑥 ∈ (suc 𝑘 ∪ {suc 𝑘})𝜑)
50		df-suc 4406	. . . . . . 7 ⊢ suc suc 𝑘 = (suc 𝑘 ∪ {suc 𝑘})
51	50	a1i 9	. . . . . 6 ⊢ ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → suc suc 𝑘 = (suc 𝑘 ∪ {suc 𝑘}))
52	51	raleqdv 2699	. . . . 5 ⊢ ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → (∀𝑥 ∈ suc suc 𝑘𝜑 ↔ ∀𝑥 ∈ (suc 𝑘 ∪ {suc 𝑘})𝜑))
53	49, 52	mpbird 167	. . . 4 ⊢ ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → ∀𝑥 ∈ suc suc 𝑘𝜑)
54	53	ex 115	. . 3 ⊢ (𝑘 ∈ ω → (∀𝑥 ∈ suc 𝑘𝜑 → ∀𝑥 ∈ suc suc 𝑘𝜑))
55	3, 5, 7, 9, 29, 54	finds 4636	. 2 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ suc 𝐴𝜑)
56		sucidg 4451	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴)
57	1, 55, 56	rspcdva 2873	1 ⊢ (𝐴 ∈ ω → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  [wsbc 2989   ∪ cun 3155  ∅c0 3450  {csn 3622  suc csuc 4400  ωcom 4626

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-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  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-uni 3840  df-int 3875  df-suc 4406  df-iom 4627

This theorem is referenced by:  nninfalllem1  15652  nninfsellemqall  15659