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Theorem tfinds3 7568
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 342 . 2 (𝑥 = ∅ → ((𝜂𝜑) ↔ (𝜂𝜓)))
3 tfinds3.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜒))
43imbi2d 342 . 2 (𝑥 = 𝑦 → ((𝜂𝜑) ↔ (𝜂𝜒)))
5 tfinds3.3 . . 3 (𝑥 = suc 𝑦 → (𝜑𝜃))
65imbi2d 342 . 2 (𝑥 = suc 𝑦 → ((𝜂𝜑) ↔ (𝜂𝜃)))
7 tfinds3.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
87imbi2d 342 . 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 3172 . . 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, 14syl5bi 243 . 2 (Lim 𝑥 → (∀𝑦𝑥 (𝜂𝜒) → (𝜂𝜑)))
162, 4, 6, 8, 9, 11, 15tfinds 7563 1 (𝐴 ∈ On → (𝜂𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  wral 3135  c0 4288  Oncon0 6184  Lim wlim 6185  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190
This theorem is referenced by:  oacl  8149  omcl  8150  oecl  8151  oawordri  8165  oaass  8176  oarec  8177  omordi  8181  omwordri  8187  odi  8194  omass  8195  oen0  8201  oewordri  8207  oeworde  8208  oeoelem  8213  omabs  8263  tfindsd  40441
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