Theorem omsinds 4623

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 4404	. . . 4 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅)
3	2	raleqdv 2679	. . 3 ⊢ (𝑤 = ∅ → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc ∅𝜑))
4		suceq 4404	. . . 4 ⊢ (𝑤 = 𝑘 → suc 𝑤 = suc 𝑘)
5	4	raleqdv 2679	. . 3 ⊢ (𝑤 = 𝑘 → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc 𝑘𝜑))
6		suceq 4404	. . . 4 ⊢ (𝑤 = suc 𝑘 → suc 𝑤 = suc suc 𝑘)
7	6	raleqdv 2679	. . 3 ⊢ (𝑤 = suc 𝑘 → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc suc 𝑘𝜑))
8		suceq 4404	. . . 4 ⊢ (𝑤 = 𝐴 → suc 𝑤 = suc 𝐴)
9	8	raleqdv 2679	. . 3 ⊢ (𝑤 = 𝐴 → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc 𝐴𝜑))
10		ral0 3526	. . . . . 6 ⊢ ∀𝑦 ∈ ∅ 𝜓
11		omsinds.3	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))
12	11	rgen 2530	. . . . . . 7 ⊢ ∀𝑥 ∈ ω (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)
13		peano1 4595	. . . . . . . 8 ⊢ ∅ ∈ ω
14	10	nfth 1464	. . . . . . . . . 10 ⊢ Ⅎ𝑥∀𝑦 ∈ ∅ 𝜓
15		nfsbc1v 2983	. . . . . . . . . 10 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑
16	14, 15	nfim 1572	. . . . . . . . 9 ⊢ Ⅎ𝑥(∀𝑦 ∈ ∅ 𝜓 → [∅ / 𝑥]𝜑)
17		raleq 2673	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓))
18		sbceq1a 2974	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑))
19	17, 18	imbi12d 234	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((∀𝑦 ∈ 𝑥 𝜓 → 𝜑) ↔ (∀𝑦 ∈ ∅ 𝜓 → [∅ / 𝑥]𝜑)))
20	16, 19	rspc 2837	. . . . . . . 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 3632	. . . . . 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 4413	. . . . 5 ⊢ suc ∅ = {∅}
28	27	raleqi 2677	. . . 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 4596	. . . . . . . . 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 2705	. . . . . . . . 9 ⊢ (∀𝑥 ∈ suc 𝑘𝜑 ↔ ∀𝑦 ∈ suc 𝑘𝜓)
35	30, 34	sylib 122	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → ∀𝑦 ∈ suc 𝑘𝜓)
36		nfv 1528	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑦 ∈ suc 𝑘𝜓
37		nfsbc1v 2983	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[suc 𝑘 / 𝑥]𝜑
38	36, 37	nfim 1572	. . . . . . . . . 10 ⊢ Ⅎ𝑥(∀𝑦 ∈ suc 𝑘𝜓 → [suc 𝑘 / 𝑥]𝜑)
39		raleq 2673	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑘 → (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑦 ∈ suc 𝑘𝜓))
40		sbceq1a 2974	. . . . . . . . . . 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 2837	. . . . . . . . 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 3632	. . . . . . . 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 3319	. . . . . 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 4373	. . . . . . 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 2679	. . . . 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 4601	. 2 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ suc 𝐴𝜑)
56		sucidg 4418	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴)
57	1, 55, 56	rspcdva 2848	1 ⊢ (𝐴 ∈ ω → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455  [wsbc 2964   ∪ cun 3129  ∅c0 3424  {csn 3594  suc csuc 4367  ωcom 4591

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 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589

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 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592

This theorem is referenced by:  nninfalllem1  14796  nninfsellemqall  14803