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Theorem tfinds3 7106
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 329 . 2 (𝑥 = ∅ → ((𝜂𝜑) ↔ (𝜂𝜓)))
3 tfinds3.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜒))
43imbi2d 329 . 2 (𝑥 = 𝑦 → ((𝜂𝜑) ↔ (𝜂𝜒)))
5 tfinds3.3 . . 3 (𝑥 = suc 𝑦 → (𝜑𝜃))
65imbi2d 329 . 2 (𝑥 = suc 𝑦 → ((𝜂𝜑) ↔ (𝜂𝜃)))
7 tfinds3.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
87imbi2d 329 . 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 2989 . . 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 232 . 2 (Lim 𝑥 → (∀𝑦𝑥 (𝜂𝜒) → (𝜂𝜑)))
162, 4, 6, 8, 9, 11, 15tfinds 7101 1 (𝐴 ∈ On → (𝜂𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wral 2941  c0 3948  Oncon0 5761  Lim wlim 5762  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767
This theorem is referenced by:  oacl  7660  omcl  7661  oecl  7662  oawordri  7675  oaass  7686  oarec  7687  omordi  7691  omwordri  7697  odi  7704  omass  7705  oen0  7711  oewordri  7717  oeworde  7718  oeoelem  7723  omabs  7772
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