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Theorem omsinds 7584
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 7568 . . 3 ω ⊆ On
2 epweon 7481 . . 3 E We On
3 wess 5510 . . 3 (ω ⊆ On → ( E We On → E We ω))
41, 2, 3mp2 9 . 2 E We ω
5 epse 5506 . 2 E Se ω
6 omsinds.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
7 omsinds.2 . 2 (𝑥 = 𝐴 → (𝜑𝜒))
8 predep 6146 . . . . 5 (𝑥 ∈ ω → Pred( E , ω, 𝑥) = (ω ∩ 𝑥))
9 ordom 7573 . . . . . . 7 Ord ω
10 ordtr 6177 . . . . . . 7 (Ord ω → Tr ω)
11 trss 5148 . . . . . . 7 (Tr ω → (𝑥 ∈ ω → 𝑥 ⊆ ω))
129, 10, 11mp2b 10 . . . . . 6 (𝑥 ∈ ω → 𝑥 ⊆ ω)
13 sseqin2 4145 . . . . . 6 (𝑥 ⊆ ω ↔ (ω ∩ 𝑥) = 𝑥)
1412, 13sylib 221 . . . . 5 (𝑥 ∈ ω → (ω ∩ 𝑥) = 𝑥)
158, 14eqtrd 2836 . . . 4 (𝑥 ∈ ω → Pred( E , ω, 𝑥) = 𝑥)
1615raleqdv 3367 . . 3 (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓 ↔ ∀𝑦𝑥 𝜓))
17 omsinds.3 . . 3 (𝑥 ∈ ω → (∀𝑦𝑥 𝜓𝜑))
1816, 17sylbid 243 . 2 (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓𝜑))
194, 5, 6, 7, 18wfis3 6161 1 (𝐴 ∈ ω → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2112  wral 3109  cin 3883  wss 3884  Tr wtr 5139   E cep 5432   We wwe 5481  Predcpred 6119  Ord word 6162  Oncon0 6163  ωcom 7564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-om 7565
This theorem is referenced by: (None)
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