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Theorem findsg 7371
 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 3846 . . . . . . 7 (𝑥 = ∅ → (𝐵𝑥𝐵 ⊆ ∅))
21adantl 475 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵𝑥𝐵 ⊆ ∅))
3 eqeq2 2789 . . . . . . . 8 (𝐵 = ∅ → (𝑥 = 𝐵𝑥 = ∅))
4 findsg.1 . . . . . . . 8 (𝑥 = 𝐵 → (𝜑𝜓))
53, 4syl6bir 246 . . . . . . 7 (𝐵 = ∅ → (𝑥 = ∅ → (𝜑𝜓)))
65imp 397 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝜑𝜓))
72, 6imbi12d 336 . . . . 5 ((𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
81imbi1d 333 . . . . . 6 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜑)))
9 ss0 4200 . . . . . . . . 9 (𝐵 ⊆ ∅ → 𝐵 = ∅)
109con3i 152 . . . . . . . 8 𝐵 = ∅ → ¬ 𝐵 ⊆ ∅)
1110pm2.21d 119 . . . . . . 7 𝐵 = ∅ → (𝐵 ⊆ ∅ → (𝜑𝜓)))
1211pm5.74d 265 . . . . . 6 𝐵 = ∅ → ((𝐵 ⊆ ∅ → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
138, 12sylan9bbr 506 . . . . 5 ((¬ 𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
147, 13pm2.61ian 802 . . . 4 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
1514imbi2d 332 . . 3 (𝑥 = ∅ → ((𝐵 ∈ ω → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ ∅ → 𝜓))))
16 sseq2 3846 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
17 findsg.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
1816, 17imbi12d 336 . . . 4 (𝑥 = 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵𝑦𝜒)))
1918imbi2d 332 . . 3 (𝑥 = 𝑦 → ((𝐵 ∈ ω → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ ω → (𝐵𝑦𝜒))))
20 sseq2 3846 . . . . 5 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ⊆ suc 𝑦))
21 findsg.3 . . . . 5 (𝑥 = suc 𝑦 → (𝜑𝜃))
2220, 21imbi12d 336 . . . 4 (𝑥 = suc 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ suc 𝑦𝜃)))
2322imbi2d 332 . . 3 (𝑥 = suc 𝑦 → ((𝐵 ∈ ω → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ suc 𝑦𝜃))))
24 sseq2 3846 . . . . 5 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
25 findsg.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
2624, 25imbi12d 336 . . . 4 (𝑥 = 𝐴 → ((𝐵𝑥𝜑) ↔ (𝐵𝐴𝜏)))
2726imbi2d 332 . . 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 3401 . . . . . . . . . . . . . 14 𝑦 ∈ V
3130sucex 7289 . . . . . . . . . . . . 13 suc 𝑦 ∈ V
3231eqvinc 3533 . . . . . . . . . . . 12 (suc 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵))
3328, 4syl5ibr 238 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐵 ∈ ω → 𝜑))
3421biimpd 221 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝜑𝜃))
3533, 34sylan9r 504 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ ω → 𝜃))
3635exlimiv 1973 . . . . . . . . . . . 12 (∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ ω → 𝜃))
3732, 36sylbi 209 . . . . . . . . . . 11 (suc 𝑦 = 𝐵 → (𝐵 ∈ ω → 𝜃))
3837eqcoms 2786 . . . . . . . . . 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 475 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
43 df-ne 2970 . . . . . . . . 9 (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
4443anbi2i 616 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ (𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦))
45 annim 394 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
4644, 45bitri 267 . . . . . . 7 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
47 nnon 7349 . . . . . . . . 9 (𝐵 ∈ ω → 𝐵 ∈ On)
48 nnon 7349 . . . . . . . . 9 (𝑦 ∈ ω → 𝑦 ∈ On)
49 onsssuc 6063 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦𝐵 ∈ suc 𝑦))
50 suceloni 7291 . . . . . . . . . . 11 (𝑦 ∈ On → suc 𝑦 ∈ On)
51 onelpss 6016 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5250, 51sylan2 586 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5349, 52bitrd 271 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5447, 48, 53syl2anr 590 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
55 findsg.6 . . . . . . . . . . . 12 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑦) → (𝜒𝜃))
5655ex 403 . . . . . . . . . . 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 252 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6146, 60syl5bir 235 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6242, 61pm2.61d 172 . . . . 5 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃)))
6362ex 403 . . . 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 7370 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐵𝐴𝜏)))
6665imp31 410 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → 𝜏)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2107   ≠ wne 2969   ⊆ wss 3792  ∅c0 4141  Oncon0 5976  suc csuc 5978  ωcom 7343 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-om 7344 This theorem is referenced by:  nnaordi  7982  inf3lem5  8826  ackbij2lem4  9399  sornom  9434  fin23lem15  9491  fin23lem36  9505  isf32lem1  9510  isf32lem2  9511  wunex2  9895  indpi  10064
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