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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 6411 . . 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 6396 . . . 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 6396 . . . . . . . . . 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 3482 . . . . . . . . . . 11 𝑥 ∈ V
28 limelon 6450 . . . . . . . . . . 11 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
2927, 28mpan 690 . . . . . . . . . 10 (Lim 𝑥𝑥 ∈ On)
30 eloni 6396 . . . . . . . . . . . 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 6411 . . . . . . . . . . . . . . . . 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 3174 . . . . . . . . . . . 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 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963  Ord word 6385  Oncon0 6386  Lim wlim 6387  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392
This theorem is referenced by:  oeordi  8624
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