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Theorem tfindsg2 6953
 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. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 5-Jan-2005.)
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 5665 . . 3 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
2 sucelon 6909 . . 3 (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
31, 2sylib 207 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → suc 𝐵 ∈ On)
4 eloni 5650 . . . 4 (𝐴 ∈ On → Ord 𝐴)
5 ordsucss 6910 . . . 4 (Ord 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
64, 5syl 17 . . 3 (𝐴 ∈ On → (𝐵𝐴 → suc 𝐵𝐴))
76imp 444 . 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 224 . . . . 5 (suc 𝐵 ∈ On → 𝜓)
14 eloni 5650 . . . . . . . . . 10 (𝑦 ∈ On → Ord 𝑦)
15 ordelsuc 6912 . . . . . . . . . 10 ((𝐵 ∈ On ∧ Ord 𝑦) → (𝐵𝑦 ↔ suc 𝐵𝑦))
1614, 15sylan2 490 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦 ↔ suc 𝐵𝑦))
1716ancoms 468 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 ↔ suc 𝐵𝑦))
18 tfindsg2.6 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝐵𝑦) → (𝜒𝜃))
1918ex 449 . . . . . . . . 9 (𝑦 ∈ On → (𝐵𝑦 → (𝜒𝜃)))
2019adantr 480 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝑦 → (𝜒𝜃)))
2117, 20sylbird 249 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐵𝑦 → (𝜒𝜃)))
222, 21sylan2br 492 . . . . . 6 ((𝑦 ∈ On ∧ suc 𝐵 ∈ On) → (suc 𝐵𝑦 → (𝜒𝜃)))
2322imp 444 . . . . 5 (((𝑦 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝑦) → (𝜒𝜃))
24 tfindsg2.7 . . . . . . . . . 10 ((Lim 𝑥𝐵𝑥) → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑))
2524ex 449 . . . . . . . . 9 (Lim 𝑥 → (𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)))
2625adantr 480 . . . . . . . 8 ((Lim 𝑥𝐵 ∈ On) → (𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)))
27 vex 3176 . . . . . . . . . . 11 𝑥 ∈ V
28 limelon 5705 . . . . . . . . . . 11 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
2927, 28mpan 702 . . . . . . . . . 10 (Lim 𝑥𝑥 ∈ On)
30 eloni 5650 . . . . . . . . . . . 12 (𝑥 ∈ On → Ord 𝑥)
31 ordelsuc 6912 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ Ord 𝑥) → (𝐵𝑥 ↔ suc 𝐵𝑥))
3230, 31sylan2 490 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵𝑥 ↔ suc 𝐵𝑥))
33 onelon 5665 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
3433, 14syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
3534, 15sylan2 490 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦𝑥)) → (𝐵𝑦 ↔ suc 𝐵𝑦))
3635anassrs 678 . . . . . . . . . . . . . 14 (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦𝑥) → (𝐵𝑦 ↔ suc 𝐵𝑦))
3736imbi1d 330 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝑥 ∈ On) ∧ 𝑦𝑥) → ((𝐵𝑦𝜒) ↔ (suc 𝐵𝑦𝜒)))
3837ralbidva 2968 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (∀𝑦𝑥 (𝐵𝑦𝜒) ↔ ∀𝑦𝑥 (suc 𝐵𝑦𝜒)))
3938imbi1d 330 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑) ↔ (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
4032, 39imbi12d 333 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4129, 40sylan2 490 . . . . . . . . 9 ((𝐵 ∈ On ∧ Lim 𝑥) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4241ancoms 468 . . . . . . . 8 ((Lim 𝑥𝐵 ∈ On) → ((𝐵𝑥 → (∀𝑦𝑥 (𝐵𝑦𝜒) → 𝜑)) ↔ (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))))
4326, 42mpbid 221 . . . . . . 7 ((Lim 𝑥𝐵 ∈ On) → (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
442, 43sylan2br 492 . . . . . 6 ((Lim 𝑥 ∧ suc 𝐵 ∈ On) → (suc 𝐵𝑥 → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑)))
4544imp 444 . . . . 5 (((Lim 𝑥 ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝑥) → (∀𝑦𝑥 (suc 𝐵𝑦𝜒) → 𝜑))
468, 9, 10, 11, 13, 23, 45tfindsg 6952 . . . 4 (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ suc 𝐵𝐴) → 𝜏)
4746expl 646 . . 3 (𝐴 ∈ On → ((suc 𝐵 ∈ On ∧ suc 𝐵𝐴) → 𝜏))
4847adantr 480 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → ((suc 𝐵 ∈ On ∧ suc 𝐵𝐴) → 𝜏))
493, 7, 48mp2and 711 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝜏)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646 This theorem is referenced by:  oeordi  7554
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