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Theorem tfinds3 7806
Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)
Hypotheses
Ref Expression
tfinds3.1 (𝑥 = ∅ → (𝜑𝜓))
tfinds3.2 (𝑥 = 𝑦 → (𝜑𝜒))
tfinds3.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
tfinds3.4 (𝑥 = 𝐴 → (𝜑𝜏))
tfinds3.5 (𝜂𝜓)
tfinds3.6 (𝑦 ∈ On → (𝜂 → (𝜒𝜃)))
tfinds3.7 (Lim 𝑥 → (𝜂 → (∀𝑦𝑥 𝜒𝜑)))
Assertion
Ref Expression
tfinds3 (𝐴 ∈ On → (𝜂𝜏))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑦   𝜒,𝑥   𝜏,𝑥   𝑥,𝑦,𝜂
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑦)   𝜃(𝑥,𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem tfinds3
StepHypRef Expression
1 tfinds3.1 . . 3 (𝑥 = ∅ → (𝜑𝜓))
21imbi2d 341 . 2 (𝑥 = ∅ → ((𝜂𝜑) ↔ (𝜂𝜓)))
3 tfinds3.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜒))
43imbi2d 341 . 2 (𝑥 = 𝑦 → ((𝜂𝜑) ↔ (𝜂𝜒)))
5 tfinds3.3 . . 3 (𝑥 = suc 𝑦 → (𝜑𝜃))
65imbi2d 341 . 2 (𝑥 = suc 𝑦 → ((𝜂𝜑) ↔ (𝜂𝜃)))
7 tfinds3.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
87imbi2d 341 . 2 (𝑥 = 𝐴 → ((𝜂𝜑) ↔ (𝜂𝜏)))
9 tfinds3.5 . 2 (𝜂𝜓)
10 tfinds3.6 . . 3 (𝑦 ∈ On → (𝜂 → (𝜒𝜃)))
1110a2d 29 . 2 (𝑦 ∈ On → ((𝜂𝜒) → (𝜂𝜃)))
12 r19.21v 3177 . . 3 (∀𝑦𝑥 (𝜂𝜒) ↔ (𝜂 → ∀𝑦𝑥 𝜒))
13 tfinds3.7 . . . 4 (Lim 𝑥 → (𝜂 → (∀𝑦𝑥 𝜒𝜑)))
1413a2d 29 . . 3 (Lim 𝑥 → ((𝜂 → ∀𝑦𝑥 𝜒) → (𝜂𝜑)))
1512, 14biimtrid 241 . 2 (Lim 𝑥 → (∀𝑦𝑥 (𝜂𝜒) → (𝜂𝜑)))
162, 4, 6, 8, 9, 11, 15tfinds 7801 1 (𝐴 ∈ On → (𝜂𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wral 3065  c0 4287  Oncon0 6322  Lim wlim 6323  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328
This theorem is referenced by:  oacl  8486  omcl  8487  oecl  8488  oawordri  8502  oaass  8513  oarec  8514  omordi  8518  omwordri  8524  odi  8531  omass  8532  oen0  8538  oewordri  8544  oeworde  8545  oeoelem  8550  omabs  8602  tfindsd  42559
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