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Theorem tfindsg2 7883
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 6409 . . 3 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
2 onsucb 7837 . . 3 (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
31, 2sylib 218 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → suc 𝐵 ∈ On)
4 eloni 6394 . . . 4 (𝐴 ∈ On → Ord 𝐴)
5 ordsucss 7838 . . . 4 (Ord 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
64, 5syl 17 . . 3 (𝐴 ∈ On → (𝐵𝐴 → suc 𝐵𝐴))
76imp 406 . 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 235 . . . . 5 (suc 𝐵 ∈ On → 𝜓)
14 eloni 6394 . . . . . . . . . 10 (𝑦 ∈ On → Ord 𝑦)
15 ordelsuc 7840 . . . . . . . . . 10 ((𝐵 ∈ On ∧ Ord 𝑦) → (𝐵𝑦 ↔ suc 𝐵𝑦))
1614, 15sylan2 593 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦 ↔ suc 𝐵𝑦))
1716ancoms 458 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 ↔ suc 𝐵𝑦))
18 tfindsg2.6 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝐵𝑦) → (𝜒𝜃))
1918ex 412 . . . . . . . . 9 (𝑦 ∈ On → (𝐵𝑦 → (𝜒𝜃)))
2019adantr 480 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → (𝜒𝜃)))
2117, 20sylbird 260 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐵𝑦 → (𝜒𝜃)))
222, 21sylan2br 595 . . . . . 6 ((𝑦 ∈ On ∧ suc 𝐵 ∈ On) → (suc 𝐵𝑦 → (𝜒𝜃)))
2322imp 406 . . . . 5 (((𝑦 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝑦) → (𝜒𝜃))
24 tfindsg2.7 . . . . . . . . . 10 ((Lim 𝑥𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))
2524ex 412 . . . . . . . . 9 (Lim 𝑥 → (𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)))
2625adantr 480 . . . . . . . 8 ((Lim 𝑥𝐵 ∈ On) → (𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)))
27 vex 3484 . . . . . . . . . . 11 𝑥 ∈ V
28 limelon 6448 . . . . . . . . . . 11 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
2927, 28mpan 690 . . . . . . . . . 10 (Lim 𝑥𝑥 ∈ On)
30 eloni 6394 . . . . . . . . . . . 12 (𝑥 ∈ On → Ord 𝑥)
31 ordelsuc 7840 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ Ord 𝑥) → (𝐵𝑥 ↔ suc 𝐵𝑥))
3230, 31sylan2 593 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵𝑥 ↔ suc 𝐵𝑥))
33 onelon 6409 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
3433, 14syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
3534, 15sylan2 593 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦𝑥)) → (𝐵𝑦 ↔ suc 𝐵𝑦))
3635anassrs 467 . . . . . . . . . . . . . 14 (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦𝑥) → (𝐵𝑦 ↔ suc 𝐵𝑦))
3736imbi1d 341 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦𝑥) → ((𝐵𝑦𝜒) ↔ (suc 𝐵𝑦𝜒)))
3837ralbidva 3176 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∀𝑦𝑥 (𝐵𝑦𝜒) ↔ ∀𝑦𝑥 (suc 𝐵𝑦𝜒)))
3938imbi1d 341 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑) ↔ (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
4032, 39imbi12d 344 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4129, 40sylan2 593 . . . . . . . . 9 ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4241ancoms 458 . . . . . . . 8 ((Lim 𝑥𝐵 ∈ On) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4326, 42mpbid 232 . . . . . . 7 ((Lim 𝑥𝐵 ∈ On) → (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
442, 43sylan2br 595 . . . . . 6 ((Lim 𝑥 ∧ suc 𝐵 ∈ On) → (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
4544imp 406 . . . . 5 (((Lim 𝑥 ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝑥) → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))
468, 9, 10, 11, 13, 23, 45tfindsg 7882 . . . 4 (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝐴) → 𝜏)
4746expl 457 . . 3 (𝐴 ∈ On → ((suc 𝐵 ∈ On ∧ suc 𝐵𝐴) → 𝜏))
4847adantr 480 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → ((suc 𝐵 ∈ On ∧ suc 𝐵𝐴) → 𝜏))
493, 7, 48mp2and 699 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390
This theorem is referenced by:  oeordi  8625
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