MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  finds Structured version   Visualization version   GIF version

Theorem finds 7889
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. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.)
Hypotheses
Ref Expression
finds.1 (𝑥 = ∅ → (𝜑𝜓))
finds.2 (𝑥 = 𝑦 → (𝜑𝜒))
finds.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
finds.4 (𝑥 = 𝐴 → (𝜑𝜏))
finds.5 𝜓
finds.6 (𝑦 ∈ ω → (𝜒𝜃))
Assertion
Ref Expression
finds (𝐴 ∈ ω → 𝜏)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   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 finds
StepHypRef Expression
1 finds.5 . . . . 5 𝜓
2 0ex 5308 . . . . . 6 ∅ ∈ V
3 finds.1 . . . . . 6 (𝑥 = ∅ → (𝜑𝜓))
42, 3elab 3669 . . . . 5 (∅ ∈ {𝑥𝜑} ↔ 𝜓)
51, 4mpbir 230 . . . 4 ∅ ∈ {𝑥𝜑}
6 finds.6 . . . . . 6 (𝑦 ∈ ω → (𝜒𝜃))
7 vex 3479 . . . . . . 7 𝑦 ∈ V
8 finds.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
97, 8elab 3669 . . . . . 6 (𝑦 ∈ {𝑥𝜑} ↔ 𝜒)
107sucex 7794 . . . . . . 7 suc 𝑦 ∈ V
11 finds.3 . . . . . . 7 (𝑥 = suc 𝑦 → (𝜑𝜃))
1210, 11elab 3669 . . . . . 6 (suc 𝑦 ∈ {𝑥𝜑} ↔ 𝜃)
136, 9, 123imtr4g 296 . . . . 5 (𝑦 ∈ ω → (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑}))
1413rgen 3064 . . . 4 𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})
15 peano5 7884 . . . 4 ((∅ ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})) → ω ⊆ {𝑥𝜑})
165, 14, 15mp2an 691 . . 3 ω ⊆ {𝑥𝜑}
1716sseli 3979 . 2 (𝐴 ∈ ω → 𝐴 ∈ {𝑥𝜑})
18 finds.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
1918elabg 3667 . 2 (𝐴 ∈ ω → (𝐴 ∈ {𝑥𝜑} ↔ 𝜏))
2017, 19mpbid 231 1 (𝐴 ∈ ω → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  {cab 2710  wral 3062  wss 3949  c0 4323  suc csuc 6367  ωcom 7855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-om 7856
This theorem is referenced by:  findsg  7890  findes  7893  seqomlem1  8450  nna0r  8609  nnm0r  8610  nnawordi  8621  nneob  8655  enrefnn  9047  pssnn  9168  nneneq  9209  nneneqOLD  9221  pssnnOLD  9265  inf3lem1  9623  inf3lem2  9624  cantnfval2  9664  cantnfp1lem3  9675  ttrclss  9715  ttrclselem2  9721  r1fin  9768  ackbij1lem14  10228  ackbij1lem16  10230  ackbij1  10233  ackbij2lem2  10235  ackbij2lem3  10236  infpssrlem4  10301  fin23lem14  10328  fin23lem34  10341  itunitc1  10415  ituniiun  10417  om2uzuzi  13914  om2uzlti  13915  om2uzrdg  13921  uzrdgxfr  13932  hashgadd  14337  mreexexd  17592  precsexlem8  27660  precsexlem9  27661  satfrel  34358  satfdm  34360  satfrnmapom  34361  satf0op  34368  satf0n0  34369  sat1el2xp  34370  fmlafvel  34376  fmlaomn0  34381  gonar  34386  goalr  34388  satffun  34400  findfvcl  35337  finxp00  36283  onmcl  42081  naddonnn  42146
  Copyright terms: Public domain W3C validator