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Theorem tfinds3 7012
 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 330 . 2 (𝑥 = ∅ → ((𝜂𝜑) ↔ (𝜂𝜓)))
3 tfinds3.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜒))
43imbi2d 330 . 2 (𝑥 = 𝑦 → ((𝜂𝜑) ↔ (𝜂𝜒)))
5 tfinds3.3 . . 3 (𝑥 = suc 𝑦 → (𝜑𝜃))
65imbi2d 330 . 2 (𝑥 = suc 𝑦 → ((𝜂𝜑) ↔ (𝜂𝜃)))
7 tfinds3.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
87imbi2d 330 . 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 2959 . . 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 7007 1 (𝐴 ∈ On → (𝜂𝜏))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1480   ∈ wcel 1992  ∀wral 2912  ∅c0 3896  Oncon0 5685  Lim wlim 5686  suc csuc 5687 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691 This theorem is referenced by:  oacl  7561  omcl  7562  oecl  7563  oawordri  7576  oaass  7587  oarec  7588  omordi  7592  omwordri  7598  odi  7605  omass  7606  oen0  7612  oewordri  7618  oeworde  7619  oeoelem  7624  omabs  7673
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