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Theorem findsg 7919
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 4010 . . . . . . 7 (𝑥 = ∅ → (𝐵𝑥𝐵 ⊆ ∅))
21adantl 481 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵𝑥𝐵 ⊆ ∅))
3 eqeq2 2749 . . . . . . . 8 (𝐵 = ∅ → (𝑥 = 𝐵𝑥 = ∅))
4 findsg.1 . . . . . . . 8 (𝑥 = 𝐵 → (𝜑𝜓))
53, 4biimtrrdi 254 . . . . . . 7 (𝐵 = ∅ → (𝑥 = ∅ → (𝜑𝜓)))
65imp 406 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝜑𝜓))
72, 6imbi12d 344 . . . . 5 ((𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
81imbi1d 341 . . . . . 6 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜑)))
9 ss0 4402 . . . . . . . . 9 (𝐵 ⊆ ∅ → 𝐵 = ∅)
109con3i 154 . . . . . . . 8 𝐵 = ∅ → ¬ 𝐵 ⊆ ∅)
1110pm2.21d 121 . . . . . . 7 𝐵 = ∅ → (𝐵 ⊆ ∅ → (𝜑𝜓)))
1211pm5.74d 273 . . . . . 6 𝐵 = ∅ → ((𝐵 ⊆ ∅ → 𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
138, 12sylan9bbr 510 . . . . 5 ((¬ 𝐵 = ∅ ∧ 𝑥 = ∅) → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
147, 13pm2.61ian 812 . . . 4 (𝑥 = ∅ → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ ∅ → 𝜓)))
1514imbi2d 340 . . 3 (𝑥 = ∅ → ((𝐵 ∈ ω → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ ∅ → 𝜓))))
16 sseq2 4010 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
17 findsg.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
1816, 17imbi12d 344 . . . 4 (𝑥 = 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵𝑦𝜒)))
1918imbi2d 340 . . 3 (𝑥 = 𝑦 → ((𝐵 ∈ ω → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ ω → (𝐵𝑦𝜒))))
20 sseq2 4010 . . . . 5 (𝑥 = suc 𝑦 → (𝐵𝑥𝐵 ⊆ suc 𝑦))
21 findsg.3 . . . . 5 (𝑥 = suc 𝑦 → (𝜑𝜃))
2220, 21imbi12d 344 . . . 4 (𝑥 = suc 𝑦 → ((𝐵𝑥𝜑) ↔ (𝐵 ⊆ suc 𝑦𝜃)))
2322imbi2d 340 . . 3 (𝑥 = suc 𝑦 → ((𝐵 ∈ ω → (𝐵𝑥𝜑)) ↔ (𝐵 ∈ ω → (𝐵 ⊆ suc 𝑦𝜃))))
24 sseq2 4010 . . . . 5 (𝑥 = 𝐴 → (𝐵𝑥𝐵𝐴))
25 findsg.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜏))
2624, 25imbi12d 344 . . . 4 (𝑥 = 𝐴 → ((𝐵𝑥𝜑) ↔ (𝐵𝐴𝜏)))
2726imbi2d 340 . . 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 3484 . . . . . . . . . . . . . 14 𝑦 ∈ V
3130sucex 7826 . . . . . . . . . . . . 13 suc 𝑦 ∈ V
3231eqvinc 3649 . . . . . . . . . . . 12 (suc 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵))
3328, 4imbitrrid 246 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝐵 ∈ ω → 𝜑))
3421biimpd 229 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → (𝜑𝜃))
3533, 34sylan9r 508 . . . . . . . . . . . . 13 ((𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ ω → 𝜃))
3635exlimiv 1930 . . . . . . . . . . . 12 (∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ ω → 𝜃))
3732, 36sylbi 217 . . . . . . . . . . 11 (suc 𝑦 = 𝐵 → (𝐵 ∈ ω → 𝜃))
3837eqcoms 2745 . . . . . . . . . 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 481 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
43 df-ne 2941 . . . . . . . . 9 (𝐵 ≠ suc 𝑦 ↔ ¬ 𝐵 = suc 𝑦)
4443anbi2i 623 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ (𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦))
45 annim 403 . . . . . . . 8 ((𝐵 ⊆ suc 𝑦 ∧ ¬ 𝐵 = suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
4644, 45bitri 275 . . . . . . 7 ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) ↔ ¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦))
47 nnon 7893 . . . . . . . . 9 (𝐵 ∈ ω → 𝐵 ∈ On)
48 nnon 7893 . . . . . . . . 9 (𝑦 ∈ ω → 𝑦 ∈ On)
49 onsssuc 6474 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦𝐵 ∈ suc 𝑦))
50 onsuc 7831 . . . . . . . . . . 11 (𝑦 ∈ On → suc 𝑦 ∈ On)
51 onelpss 6424 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ suc 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5250, 51sylan2 593 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ∈ suc 𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5349, 52bitrd 279 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
5447, 48, 53syl2anr 597 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵𝑦 ↔ (𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦)))
55 findsg.6 . . . . . . . . . . . 12 (((𝑦 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑦) → (𝜒𝜃))
5655ex 412 . . . . . . . . . . 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 260 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵 ⊆ suc 𝑦𝐵 ≠ suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6146, 60biimtrrid 243 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (¬ (𝐵 ⊆ suc 𝑦𝐵 = suc 𝑦) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃))))
6242, 61pm2.61d 179 . . . . 5 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐵𝑦𝜒) → (𝐵 ⊆ suc 𝑦𝜃)))
6362ex 412 . . . 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 7918 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐵𝐴𝜏)))
6665imp31 417 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wss 3951  c0 4333  Oncon0 6384  suc csuc 6386  ωcom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-om 7888
This theorem is referenced by:  nnaordi  8656  inf3lem5  9672  ackbij2lem4  10281  sornom  10317  fin23lem15  10374  fin23lem36  10388  isf32lem1  10393  isf32lem2  10394  wunex2  10778  indpi  10947  satfsschain  35369
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