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Theorem tfindsg2 7569
 Description: 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. The basis of this version is an arbitrary ordinal suc 𝐵 instead of zero. (Contributed by NM, 5-Jan-2005.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
tfindsg2.1 (𝑥 = suc 𝐵 → (𝜑𝜓))
tfindsg2.2 (𝑥 = 𝑦 → (𝜑𝜒))
tfindsg2.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
tfindsg2.4 (𝑥 = 𝐴 → (𝜑𝜏))
tfindsg2.5 (𝐵 ∈ On → 𝜓)
tfindsg2.6 ((𝑦 ∈ On ∧ 𝐵𝑦) → (𝜒𝜃))
tfindsg2.7 ((Lim 𝑥𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))
Assertion
Ref Expression
tfindsg2 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝜏)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem tfindsg2
StepHypRef Expression
1 onelon 6191 . . 3 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
2 sucelon 7525 . . 3 (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
31, 2sylib 221 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → suc 𝐵 ∈ On)
4 eloni 6176 . . . 4 (𝐴 ∈ On → Ord 𝐴)
5 ordsucss 7526 . . . 4 (Ord 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
64, 5syl 17 . . 3 (𝐴 ∈ On → (𝐵𝐴 → suc 𝐵𝐴))
76imp 410 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → suc 𝐵𝐴)
8 tfindsg2.1 . . . . 5 (𝑥 = suc 𝐵 → (𝜑𝜓))
9 tfindsg2.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
10 tfindsg2.3 . . . . 5 (𝑥 = suc 𝑦 → (𝜑𝜃))
11 tfindsg2.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
12 tfindsg2.5 . . . . . 6 (𝐵 ∈ On → 𝜓)
132, 12sylbir 238 . . . . 5 (suc 𝐵 ∈ On → 𝜓)
14 eloni 6176 . . . . . . . . . 10 (𝑦 ∈ On → Ord 𝑦)
15 ordelsuc 7528 . . . . . . . . . 10 ((𝐵 ∈ On ∧ Ord 𝑦) → (𝐵𝑦 ↔ suc 𝐵𝑦))
1614, 15sylan2 595 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦 ↔ suc 𝐵𝑦))
1716ancoms 462 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 ↔ suc 𝐵𝑦))
18 tfindsg2.6 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝐵𝑦) → (𝜒𝜃))
1918ex 416 . . . . . . . . 9 (𝑦 ∈ On → (𝐵𝑦 → (𝜒𝜃)))
2019adantr 484 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → (𝜒𝜃)))
2117, 20sylbird 263 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐵𝑦 → (𝜒𝜃)))
222, 21sylan2br 597 . . . . . 6 ((𝑦 ∈ On ∧ suc 𝐵 ∈ On) → (suc 𝐵𝑦 → (𝜒𝜃)))
2322imp 410 . . . . 5 (((𝑦 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝑦) → (𝜒𝜃))
24 tfindsg2.7 . . . . . . . . . 10 ((Lim 𝑥𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))
2524ex 416 . . . . . . . . 9 (Lim 𝑥 → (𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)))
2625adantr 484 . . . . . . . 8 ((Lim 𝑥𝐵 ∈ On) → (𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)))
27 vex 3445 . . . . . . . . . . 11 𝑥 ∈ V
28 limelon 6229 . . . . . . . . . . 11 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
2927, 28mpan 689 . . . . . . . . . 10 (Lim 𝑥𝑥 ∈ On)
30 eloni 6176 . . . . . . . . . . . 12 (𝑥 ∈ On → Ord 𝑥)
31 ordelsuc 7528 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ Ord 𝑥) → (𝐵𝑥 ↔ suc 𝐵𝑥))
3230, 31sylan2 595 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵𝑥 ↔ suc 𝐵𝑥))
33 onelon 6191 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
3433, 14syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
3534, 15sylan2 595 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦𝑥)) → (𝐵𝑦 ↔ suc 𝐵𝑦))
3635anassrs 471 . . . . . . . . . . . . . 14 (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦𝑥) → (𝐵𝑦 ↔ suc 𝐵𝑦))
3736imbi1d 345 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦𝑥) → ((𝐵𝑦𝜒) ↔ (suc 𝐵𝑦𝜒)))
3837ralbidva 3161 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∀𝑦𝑥 (𝐵𝑦𝜒) ↔ ∀𝑦𝑥 (suc 𝐵𝑦𝜒)))
3938imbi1d 345 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑) ↔ (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
4032, 39imbi12d 348 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4129, 40sylan2 595 . . . . . . . . 9 ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4241ancoms 462 . . . . . . . 8 ((Lim 𝑥𝐵 ∈ On) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4326, 42mpbid 235 . . . . . . 7 ((Lim 𝑥𝐵 ∈ On) → (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
442, 43sylan2br 597 . . . . . 6 ((Lim 𝑥 ∧ suc 𝐵 ∈ On) → (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
4544imp 410 . . . . 5 (((Lim 𝑥 ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝑥) → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))
468, 9, 10, 11, 13, 23, 45tfindsg 7568 . . . 4 (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝐴) → 𝜏)
4746expl 461 . . 3 (𝐴 ∈ On → ((suc 𝐵 ∈ On ∧ suc 𝐵𝐴) → 𝜏))
4847adantr 484 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → ((suc 𝐵 ∈ On ∧ suc 𝐵𝐴) → 𝜏))
493, 7, 48mp2and 698 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝜏)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3442   ⊆ wss 3883  Ord word 6165  Oncon0 6166  Lim wlim 6167  suc csuc 6168 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-tr 5141  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172 This theorem is referenced by:  oeordi  8214
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