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Theorem tfindsg2 7565
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 6209 . . 3 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
2 sucelon 7521 . . 3 (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
31, 2sylib 219 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → suc 𝐵 ∈ On)
4 eloni 6194 . . . 4 (𝐴 ∈ On → Ord 𝐴)
5 ordsucss 7522 . . . 4 (Ord 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
64, 5syl 17 . . 3 (𝐴 ∈ On → (𝐵𝐴 → suc 𝐵𝐴))
76imp 407 . 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 236 . . . . 5 (suc 𝐵 ∈ On → 𝜓)
14 eloni 6194 . . . . . . . . . 10 (𝑦 ∈ On → Ord 𝑦)
15 ordelsuc 7524 . . . . . . . . . 10 ((𝐵 ∈ On ∧ Ord 𝑦) → (𝐵𝑦 ↔ suc 𝐵𝑦))
1614, 15sylan2 592 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦 ↔ suc 𝐵𝑦))
1716ancoms 459 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 ↔ suc 𝐵𝑦))
18 tfindsg2.6 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝐵𝑦) → (𝜒𝜃))
1918ex 413 . . . . . . . . 9 (𝑦 ∈ On → (𝐵𝑦 → (𝜒𝜃)))
2019adantr 481 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → (𝜒𝜃)))
2117, 20sylbird 261 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐵𝑦 → (𝜒𝜃)))
222, 21sylan2br 594 . . . . . 6 ((𝑦 ∈ On ∧ suc 𝐵 ∈ On) → (suc 𝐵𝑦 → (𝜒𝜃)))
2322imp 407 . . . . 5 (((𝑦 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝑦) → (𝜒𝜃))
24 tfindsg2.7 . . . . . . . . . 10 ((Lim 𝑥𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))
2524ex 413 . . . . . . . . 9 (Lim 𝑥 → (𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)))
2625adantr 481 . . . . . . . 8 ((Lim 𝑥𝐵 ∈ On) → (𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)))
27 vex 3495 . . . . . . . . . . 11 𝑥 ∈ V
28 limelon 6247 . . . . . . . . . . 11 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
2927, 28mpan 686 . . . . . . . . . 10 (Lim 𝑥𝑥 ∈ On)
30 eloni 6194 . . . . . . . . . . . 12 (𝑥 ∈ On → Ord 𝑥)
31 ordelsuc 7524 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ Ord 𝑥) → (𝐵𝑥 ↔ suc 𝐵𝑥))
3230, 31sylan2 592 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵𝑥 ↔ suc 𝐵𝑥))
33 onelon 6209 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
3433, 14syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
3534, 15sylan2 592 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦𝑥)) → (𝐵𝑦 ↔ suc 𝐵𝑦))
3635anassrs 468 . . . . . . . . . . . . . 14 (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦𝑥) → (𝐵𝑦 ↔ suc 𝐵𝑦))
3736imbi1d 343 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦𝑥) → ((𝐵𝑦𝜒) ↔ (suc 𝐵𝑦𝜒)))
3837ralbidva 3193 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∀𝑦𝑥 (𝐵𝑦𝜒) ↔ ∀𝑦𝑥 (suc 𝐵𝑦𝜒)))
3938imbi1d 343 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑) ↔ (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
4032, 39imbi12d 346 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4129, 40sylan2 592 . . . . . . . . 9 ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4241ancoms 459 . . . . . . . 8 ((Lim 𝑥𝐵 ∈ On) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4326, 42mpbid 233 . . . . . . 7 ((Lim 𝑥𝐵 ∈ On) → (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
442, 43sylan2br 594 . . . . . 6 ((Lim 𝑥 ∧ suc 𝐵 ∈ On) → (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
4544imp 407 . . . . 5 (((Lim 𝑥 ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝑥) → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))
468, 9, 10, 11, 13, 23, 45tfindsg 7564 . . . 4 (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝐴) → 𝜏)
4746expl 458 . . 3 (𝐴 ∈ On → ((suc 𝐵 ∈ On ∧ suc 𝐵𝐴) → 𝜏))
4847adantr 481 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → ((suc 𝐵 ∈ On ∧ suc 𝐵𝐴) → 𝜏))
493, 7, 48mp2and 695 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  wss 3933  Ord word 6183  Oncon0 6184  Lim wlim 6185  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190
This theorem is referenced by:  oeordi  8202
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