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Theorem omsinds 7046
Description: Strong (or "total") induction principle over the finite ordinals. (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
StepHypRef Expression
1 omsson 7031 . . 3 ω ⊆ On
2 epweon 6945 . . 3 E We On
3 wess 5071 . . 3 (ω ⊆ On → ( E We On → E We ω))
41, 2, 3mp2 9 . 2 E We ω
5 epse 5067 . 2 E Se ω
6 omsinds.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
7 omsinds.2 . 2 (𝑥 = 𝐴 → (𝜑𝜒))
8 predep 5675 . . . . 5 (𝑥 ∈ ω → Pred( E , ω, 𝑥) = (ω ∩ 𝑥))
9 ordom 7036 . . . . . . 7 Ord ω
10 ordtr 5706 . . . . . . 7 (Ord ω → Tr ω)
11 trss 4731 . . . . . . 7 (Tr ω → (𝑥 ∈ ω → 𝑥 ⊆ ω))
129, 10, 11mp2b 10 . . . . . 6 (𝑥 ∈ ω → 𝑥 ⊆ ω)
13 sseqin2 3801 . . . . . 6 (𝑥 ⊆ ω ↔ (ω ∩ 𝑥) = 𝑥)
1412, 13sylib 208 . . . . 5 (𝑥 ∈ ω → (ω ∩ 𝑥) = 𝑥)
158, 14eqtrd 2655 . . . 4 (𝑥 ∈ ω → Pred( E , ω, 𝑥) = 𝑥)
1615raleqdv 3137 . . 3 (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓 ↔ ∀𝑦𝑥 𝜓))
17 omsinds.3 . . 3 (𝑥 ∈ ω → (∀𝑦𝑥 𝜓𝜑))
1816, 17sylbid 230 . 2 (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓𝜑))
194, 5, 6, 7, 18wfis3 5690 1 (𝐴 ∈ ω → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  wral 2908  cin 3559  wss 3560  Tr wtr 4722   E cep 4993   We wwe 5042  Predcpred 5648  Ord word 5691  Oncon0 5692  ωcom 7027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-om 7028
This theorem is referenced by: (None)
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