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Theorem omsinds 7895
Description: Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.) (Proof shortened by BJ, 16-Oct-2024.)
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 7878 . . 3 ω ⊆ On
2 epweon 7781 . . 3 E We On
3 wess 5667 . . 3 (ω ⊆ On → ( E We On → E We ω))
41, 2, 3mp2 9 . 2 E We ω
5 epse 5663 . 2 E Se ω
6 omsinds.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
7 omsinds.2 . 2 (𝑥 = 𝐴 → (𝜑𝜒))
8 trom 7883 . . . . 5 Tr ω
9 trpred 6340 . . . . 5 ((Tr ω ∧ 𝑥 ∈ ω) → Pred( E , ω, 𝑥) = 𝑥)
108, 9mpan 688 . . . 4 (𝑥 ∈ ω → Pred( E , ω, 𝑥) = 𝑥)
1110raleqdv 3321 . . 3 (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓 ↔ ∀𝑦𝑥 𝜓))
12 omsinds.3 . . 3 (𝑥 ∈ ω → (∀𝑦𝑥 𝜓𝜑))
1311, 12sylbid 239 . 2 (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓𝜑))
144, 5, 6, 7, 13wfis3 6370 1 (𝐴 ∈ ω → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  wral 3057  wss 3947  Tr wtr 5267   E cep 5583   We wwe 5634  Predcpred 6307  Oncon0 6372  ωcom 7874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-tr 5268  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-se 5636  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-om 7875
This theorem is referenced by: (None)
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