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Theorem findsg 7920
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 4022 . . . . . . 7 (𝑥 = ∅ → (𝐵𝑥𝐵 ⊆ ∅))
21adantl 481 . . . . . 6 ((𝐵 = ∅ ∧ 𝑥 = ∅) → (𝐵𝑥𝐵 ⊆ ∅))
3 eqeq2 2747 . . . . . . . 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 4408 . . . . . . . . 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 4022 . . . . 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 4022 . . . . 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 4022 . . . . 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 3482 . . . . . . . . . . . . . 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 1928 . . . . . . . . . . . 12 (∃𝑥(𝑥 = suc 𝑦𝑥 = 𝐵) → (𝐵 ∈ ω → 𝜃))
3732, 36sylbi 217 . . . . . . . . . . 11 (suc 𝑦 = 𝐵 → (𝐵 ∈ ω → 𝜃))
3837eqcoms 2743 . . . . . . . . . 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 2939 . . . . . . . . 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 6476 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵𝑦𝐵 ∈ suc 𝑦))
50 onsuc 7831 . . . . . . . . . . 11 (𝑦 ∈ On → suc 𝑦 ∈ On)
51 onelpss 6426 . . . . . . . . . . 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 7919 . 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 1537  wex 1776  wcel 2106  wne 2938  wss 3963  c0 4339  Oncon0 6386  suc csuc 6388  ωcom 7887
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-om 7888
This theorem is referenced by:  nnaordi  8655  inf3lem5  9670  ackbij2lem4  10279  sornom  10315  fin23lem15  10372  fin23lem36  10386  isf32lem1  10391  isf32lem2  10392  wunex2  10776  indpi  10945  satfsschain  35349
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