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Theorem omsinds 4425
Description: Strong (or "total") induction principle over ω. (Contributed by Scott Fenton, 17-Jul-2015.)
Hypotheses
Ref Expression
omsinds.1 (𝑥 = 𝑦 → (𝜑𝜓))
omsinds.2 (𝑥 = 𝐴 → (𝜑𝜒))
omsinds.3 (𝑥 ∈ ω → (∀𝑦𝑥 𝜓𝜑))
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.
StepHypRef Expression
1 omsinds.2 . 2 (𝑥 = 𝐴 → (𝜑𝜒))
2 suceq 4220 . . . 4 (𝑤 = ∅ → suc 𝑤 = suc ∅)
32raleqdv 2568 . . 3 (𝑤 = ∅ → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc ∅𝜑))
4 suceq 4220 . . . 4 (𝑤 = 𝑘 → suc 𝑤 = suc 𝑘)
54raleqdv 2568 . . 3 (𝑤 = 𝑘 → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc 𝑘𝜑))
6 suceq 4220 . . . 4 (𝑤 = suc 𝑘 → suc 𝑤 = suc suc 𝑘)
76raleqdv 2568 . . 3 (𝑤 = suc 𝑘 → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc suc 𝑘𝜑))
8 suceq 4220 . . . 4 (𝑤 = 𝐴 → suc 𝑤 = suc 𝐴)
98raleqdv 2568 . . 3 (𝑤 = 𝐴 → (∀𝑥 ∈ suc 𝑤𝜑 ↔ ∀𝑥 ∈ suc 𝐴𝜑))
10 ral0 3379 . . . . . 6 𝑦 ∈ ∅ 𝜓
11 omsinds.3 . . . . . . . 8 (𝑥 ∈ ω → (∀𝑦𝑥 𝜓𝜑))
1211rgen 2428 . . . . . . 7 𝑥 ∈ ω (∀𝑦𝑥 𝜓𝜑)
13 peano1 4399 . . . . . . . 8 ∅ ∈ ω
1410nfth 1398 . . . . . . . . . 10 𝑥𝑦 ∈ ∅ 𝜓
15 nfsbc1v 2856 . . . . . . . . . 10 𝑥[∅ / 𝑥]𝜑
1614, 15nfim 1509 . . . . . . . . 9 𝑥(∀𝑦 ∈ ∅ 𝜓[∅ / 𝑥]𝜑)
17 raleq 2562 . . . . . . . . . 10 (𝑥 = ∅ → (∀𝑦𝑥 𝜓 ↔ ∀𝑦 ∈ ∅ 𝜓))
18 sbceq1a 2847 . . . . . . . . . 10 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
1917, 18imbi12d 232 . . . . . . . . 9 (𝑥 = ∅ → ((∀𝑦𝑥 𝜓𝜑) ↔ (∀𝑦 ∈ ∅ 𝜓[∅ / 𝑥]𝜑)))
2016, 19rspc 2716 . . . . . . . 8 (∅ ∈ ω → (∀𝑥 ∈ ω (∀𝑦𝑥 𝜓𝜑) → (∀𝑦 ∈ ∅ 𝜓[∅ / 𝑥]𝜑)))
2113, 20ax-mp 7 . . . . . . 7 (∀𝑥 ∈ ω (∀𝑦𝑥 𝜓𝜑) → (∀𝑦 ∈ ∅ 𝜓[∅ / 𝑥]𝜑))
2212, 21ax-mp 7 . . . . . 6 (∀𝑦 ∈ ∅ 𝜓[∅ / 𝑥]𝜑)
2310, 22ax-mp 7 . . . . 5 [∅ / 𝑥]𝜑
24 ralsns 3476 . . . . . 6 (∅ ∈ ω → (∀𝑥 ∈ {∅}𝜑[∅ / 𝑥]𝜑))
2513, 24ax-mp 7 . . . . 5 (∀𝑥 ∈ {∅}𝜑[∅ / 𝑥]𝜑)
2623, 25mpbir 144 . . . 4 𝑥 ∈ {∅}𝜑
27 suc0 4229 . . . . 5 suc ∅ = {∅}
2827raleqi 2566 . . . 4 (∀𝑥 ∈ suc ∅𝜑 ↔ ∀𝑥 ∈ {∅}𝜑)
2926, 28mpbir 144 . . 3 𝑥 ∈ suc ∅𝜑
30 simpr 108 . . . . . 6 ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → ∀𝑥 ∈ suc 𝑘𝜑)
31 peano2 4400 . . . . . . . . 9 (𝑘 ∈ ω → suc 𝑘 ∈ ω)
3231adantr 270 . . . . . . . 8 ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → suc 𝑘 ∈ ω)
33 omsinds.1 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜓))
3433cbvralv 2590 . . . . . . . . 9 (∀𝑥 ∈ suc 𝑘𝜑 ↔ ∀𝑦 ∈ suc 𝑘𝜓)
3530, 34sylib 120 . . . . . . . 8 ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → ∀𝑦 ∈ suc 𝑘𝜓)
36 nfv 1466 . . . . . . . . . . 11 𝑥𝑦 ∈ suc 𝑘𝜓
37 nfsbc1v 2856 . . . . . . . . . . 11 𝑥[suc 𝑘 / 𝑥]𝜑
3836, 37nfim 1509 . . . . . . . . . 10 𝑥(∀𝑦 ∈ suc 𝑘𝜓[suc 𝑘 / 𝑥]𝜑)
39 raleq 2562 . . . . . . . . . . 11 (𝑥 = suc 𝑘 → (∀𝑦𝑥 𝜓 ↔ ∀𝑦 ∈ suc 𝑘𝜓))
40 sbceq1a 2847 . . . . . . . . . . 11 (𝑥 = suc 𝑘 → (𝜑[suc 𝑘 / 𝑥]𝜑))
4139, 40imbi12d 232 . . . . . . . . . 10 (𝑥 = suc 𝑘 → ((∀𝑦𝑥 𝜓𝜑) ↔ (∀𝑦 ∈ suc 𝑘𝜓[suc 𝑘 / 𝑥]𝜑)))
4238, 41rspc 2716 . . . . . . . . 9 (suc 𝑘 ∈ ω → (∀𝑥 ∈ ω (∀𝑦𝑥 𝜓𝜑) → (∀𝑦 ∈ suc 𝑘𝜓[suc 𝑘 / 𝑥]𝜑)))
4312, 42mpi 15 . . . . . . . 8 (suc 𝑘 ∈ ω → (∀𝑦 ∈ suc 𝑘𝜓[suc 𝑘 / 𝑥]𝜑))
4432, 35, 43sylc 61 . . . . . . 7 ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → [suc 𝑘 / 𝑥]𝜑)
45 ralsns 3476 . . . . . . . 8 (suc 𝑘 ∈ ω → (∀𝑥 ∈ {suc 𝑘}𝜑[suc 𝑘 / 𝑥]𝜑))
4632, 45syl 14 . . . . . . 7 ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → (∀𝑥 ∈ {suc 𝑘}𝜑[suc 𝑘 / 𝑥]𝜑))
4744, 46mpbird 165 . . . . . 6 ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → ∀𝑥 ∈ {suc 𝑘}𝜑)
48 ralun 3180 . . . . . 6 ((∀𝑥 ∈ suc 𝑘𝜑 ∧ ∀𝑥 ∈ {suc 𝑘}𝜑) → ∀𝑥 ∈ (suc 𝑘 ∪ {suc 𝑘})𝜑)
4930, 47, 48syl2anc 403 . . . . 5 ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → ∀𝑥 ∈ (suc 𝑘 ∪ {suc 𝑘})𝜑)
50 df-suc 4189 . . . . . . 7 suc suc 𝑘 = (suc 𝑘 ∪ {suc 𝑘})
5150a1i 9 . . . . . 6 ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → suc suc 𝑘 = (suc 𝑘 ∪ {suc 𝑘}))
5251raleqdv 2568 . . . . 5 ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → (∀𝑥 ∈ suc suc 𝑘𝜑 ↔ ∀𝑥 ∈ (suc 𝑘 ∪ {suc 𝑘})𝜑))
5349, 52mpbird 165 . . . 4 ((𝑘 ∈ ω ∧ ∀𝑥 ∈ suc 𝑘𝜑) → ∀𝑥 ∈ suc suc 𝑘𝜑)
5453ex 113 . . 3 (𝑘 ∈ ω → (∀𝑥 ∈ suc 𝑘𝜑 → ∀𝑥 ∈ suc suc 𝑘𝜑))
553, 5, 7, 9, 29, 54finds 4405 . 2 (𝐴 ∈ ω → ∀𝑥 ∈ suc 𝐴𝜑)
56 sucidg 4234 . 2 (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴)
571, 55, 56rspcdva 2727 1 (𝐴 ∈ ω → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359  [wsbc 2838  cun 2995  c0 3284  {csn 3441  suc csuc 4183  ωcom 4395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-int 3684  df-suc 4189  df-iom 4396
This theorem is referenced by:  nninfalllem1  11545  nninfsellemqall  11553
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