MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  finds2 Structured version   Visualization version   GIF version

Theorem finds2 7920
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
Hypotheses
Ref Expression
finds2.1 (𝑥 = ∅ → (𝜑𝜓))
finds2.2 (𝑥 = 𝑦 → (𝜑𝜒))
finds2.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
finds2.4 (𝜏𝜓)
finds2.5 (𝑦 ∈ ω → (𝜏 → (𝜒𝜃)))
Assertion
Ref Expression
finds2 (𝑥 ∈ ω → (𝜏𝜑))
Distinct variable groups:   𝑥,𝑦,𝜏   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)

Proof of Theorem finds2
StepHypRef Expression
1 finds2.4 . . . . 5 (𝜏𝜓)
2 0ex 5307 . . . . . 6 ∅ ∈ V
3 finds2.1 . . . . . . 7 (𝑥 = ∅ → (𝜑𝜓))
43imbi2d 340 . . . . . 6 (𝑥 = ∅ → ((𝜏𝜑) ↔ (𝜏𝜓)))
52, 4elab 3679 . . . . 5 (∅ ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜓))
61, 5mpbir 231 . . . 4 ∅ ∈ {𝑥 ∣ (𝜏𝜑)}
7 finds2.5 . . . . . . 7 (𝑦 ∈ ω → (𝜏 → (𝜒𝜃)))
87a2d 29 . . . . . 6 (𝑦 ∈ ω → ((𝜏𝜒) → (𝜏𝜃)))
9 vex 3484 . . . . . . 7 𝑦 ∈ V
10 finds2.2 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜒))
1110imbi2d 340 . . . . . . 7 (𝑥 = 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜒)))
129, 11elab 3679 . . . . . 6 (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜒))
139sucex 7826 . . . . . . 7 suc 𝑦 ∈ V
14 finds2.3 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝜑𝜃))
1514imbi2d 340 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜃)))
1613, 15elab 3679 . . . . . 6 (suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜃))
178, 12, 163imtr4g 296 . . . . 5 (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)}))
1817rgen 3063 . . . 4 𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)})
19 peano5 7915 . . . 4 ((∅ ∈ {𝑥 ∣ (𝜏𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏𝜑)})
206, 18, 19mp2an 692 . . 3 ω ⊆ {𝑥 ∣ (𝜏𝜑)}
2120sseli 3979 . 2 (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏𝜑)})
22 abid 2718 . 2 (𝑥 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜑))
2321, 22sylib 218 1 (𝑥 ∈ ω → (𝜏𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wss 3951  c0 4333  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-12 2177  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:  finds1  7921  onnseq  8384  nnacl  8649  nnmcl  8650  nnecl  8651  nnacom  8655  nnaass  8660  nndi  8661  nnmass  8662  nnmsucr  8663  nnmcom  8664  nnmordi  8669  omsmolem  8695  isinf  9296  isinfOLD  9297  unblem2  9329  fiint  9366  fiintOLD  9367  dffi3  9471  card2inf  9595  cantnfle  9711  cantnflt  9712  cantnflem1  9729  cnfcom  9740  trcl  9768  fseqenlem1  10064  nnadju  10238  infpssrlem3  10345  fin23lem26  10365  axdc3lem2  10491  axdc4lem  10495  axdclem2  10560  wunr1om  10759  wuncval2  10787  tskr1om  10807  grothomex  10869  peano5nni  12269  precsexlem6  28236  precsexlem7  28237  noseqind  28298  om2noseqlt  28305  neibastop2lem  36361  finxpreclem6  37397  domalom  37405  oaabsb  43307
  Copyright terms: Public domain W3C validator