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Theorem findsg 7608
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. The basis of this version is an arbitrary natural number 𝐵 instead of zero. (Contributed by NM, 16-Sep-1995.)
Hypotheses
Ref Expression
findsg.1 (𝑥 = 𝐵 → (𝜑𝜓))
findsg.2 (𝑥 = 𝑦 → (𝜑𝜒))
findsg.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
findsg.4 (𝑥 = 𝐴 → (𝜑𝜏))
findsg.5 (𝐵 ∈ ω → 𝜓)
findsg.6 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑦) → (𝜒𝜃))
Assertion
Ref Expression
findsg (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → 𝜏)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem findsg
StepHypRef Expression
1 sseq2 3992 . . . . . . 7 (𝑥 = ∅ → (𝐵𝑥𝐵 ⊆ ∅))
21adantl 484 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵𝑥𝐵 ⊆ ∅))
3 eqeq2 2833 . . . . . . . 8 (𝐵 = ∅ → (𝑥 = 𝐵𝑥 = ∅))
4 findsg.1 . . . . . . . 8 (𝑥 = 𝐵 → (𝜑𝜓))
53, 4syl6bir 256 . . . . . . 7 (𝐵 = ∅ → (𝑥 = ∅ → (𝜑𝜓)))
65imp 409 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝜑𝜓))
72, 6imbi12d 347 . . . . 5 ((𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
81imbi1d 344 . . . . . 6 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜑)))
9 ss0 4351 . . . . . . . . 9 (𝐵 ⊆ ∅ → 𝐵 = ∅)
109con3i 157 . . . . . . . 8 𝐵 = ∅ → ¬ 𝐵 ⊆ ∅)
1110pm2.21d 121 . . . . . . 7 𝐵 = ∅ → (𝐵 ⊆ ∅ → (𝜑𝜓)))
1211pm5.74d 275 . . . . . 6 𝐵 = ∅ → ((𝐵 ⊆ ∅ → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
138, 12sylan9bbr 513 . . . . 5 ((¬ 𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
147, 13pm2.61ian 810 . . . 4 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
1514imbi2d 343 . . 3 (𝑥 = ∅ → ((𝐵 ∈ ω → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ ∅ → 𝜓))))
16 sseq2 3992 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
17 findsg.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
1816, 17imbi12d 347 . . . 4 (𝑥 = 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵𝑦𝜒)))
1918imbi2d 343 . . 3 (𝑥 = 𝑦 → ((𝐵 ∈ ω → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ ω → (𝐵𝑦𝜒))))
20 sseq2 3992 . . . . 5 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ⊆ suc 𝑦))
21 findsg.3 . . . . 5 (𝑥 = suc 𝑦 → (𝜑𝜃))
2220, 21imbi12d 347 . . . 4 (𝑥 = suc 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ suc 𝑦𝜃)))
2322imbi2d 343 . . 3 (𝑥 = suc 𝑦 → ((𝐵 ∈ ω → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ suc 𝑦𝜃))))
24 sseq2 3992 . . . . 5 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
25 findsg.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
2624, 25imbi12d 347 . . . 4 (𝑥 = 𝐴 → ((𝐵𝑥𝜑) ↔ (𝐵𝐴𝜏)))
2726imbi2d 343 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ ω → (𝐵𝐴𝜏))))
28 findsg.5 . . . 4 (𝐵 ∈ ω → 𝜓)
2928a1d 25 . . 3 (𝐵 ∈ ω → (𝐵 ⊆ ∅ → 𝜓))
30 vex 3497 . . . . . . . . . . . . . 14 𝑦 ∈ V
3130sucex 7525 . . . . . . . . . . . . 13 suc 𝑦 ∈ V
3231eqvinc 3641 . . . . . . . . . . . 12 (suc 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵))
3328, 4syl5ibr 248 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐵 ∈ ω → 𝜑))
3421biimpd 231 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝜑𝜃))
3533, 34sylan9r 511 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ ω → 𝜃))
3635exlimiv 1927 . . . . . . . . . . . 12 (∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ ω → 𝜃))
3732, 36sylbi 219 . . . . . . . . . . 11 (suc 𝑦 = 𝐵 → (𝐵 ∈ ω → 𝜃))
3837eqcoms 2829 . . . . . . . . . 10 (𝐵 = suc 𝑦 → (𝐵 ∈ ω → 𝜃))
3938imim2i 16 . . . . . . . . 9 ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → (𝐵 ⊆ suc 𝑦 → (𝐵 ∈ ω → 𝜃)))
4039a1d 25 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦 → (𝐵 ∈ ω → 𝜃))))
4140com4r 94 . . . . . . 7 (𝐵 ∈ ω → ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
4241adantl 484 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
43 df-ne 3017 . . . . . . . . 9 (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
4443anbi2i 624 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ (𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦))
45 annim 406 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
4644, 45bitri 277 . . . . . . 7 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
47 nnon 7585 . . . . . . . . 9 (𝐵 ∈ ω → 𝐵 ∈ On)
48 nnon 7585 . . . . . . . . 9 (𝑦 ∈ ω → 𝑦 ∈ On)
49 onsssuc 6277 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦𝐵 ∈ suc 𝑦))
50 suceloni 7527 . . . . . . . . . . 11 (𝑦 ∈ On → suc 𝑦 ∈ On)
51 onelpss 6230 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5250, 51sylan2 594 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5349, 52bitrd 281 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5447, 48, 53syl2anr 598 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
55 findsg.6 . . . . . . . . . . . 12 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑦) → (𝜒𝜃))
5655ex 415 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵𝑦 → (𝜒𝜃)))
5756a1ddd 80 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵𝑦 → (𝜒 → (𝐵 ⊆ suc 𝑦𝜃))))
5857a2d 29 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵𝑦𝜒) → (𝐵𝑦 → (𝐵 ⊆ suc 𝑦𝜃))))
5958com23 86 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵𝑦 → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6054, 59sylbird 262 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6146, 60syl5bir 245 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6242, 61pm2.61d 181 . . . . 5 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃)))
6362ex 415 . . . 4 (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6463a2d 29 . . 3 (𝑦 ∈ ω → ((𝐵 ∈ ω → (𝐵𝑦𝜒)) → (𝐵 ∈ ω → (𝐵 ⊆ suc 𝑦𝜃))))
6515, 19, 23, 27, 29, 64finds 7607 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐵𝐴𝜏)))
6665imp31 420 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1533  wex 1776  wcel 2110  wne 3016  wss 3935  c0 4290  Oncon0 6190  suc csuc 6192  ωcom 7579
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-tr 5172  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-om 7580
This theorem is referenced by:  nnaordi  8243  inf3lem5  9094  ackbij2lem4  9663  sornom  9698  fin23lem15  9755  fin23lem36  9769  isf32lem1  9774  isf32lem2  9775  wunex2  10159  indpi  10328  satfsschain  32611
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