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Theorem finds2 7594
 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 5176 . . . . . 6 ∅ ∈ V
3 finds2.1 . . . . . . 7 (𝑥 = ∅ → (𝜑𝜓))
43imbi2d 344 . . . . . 6 (𝑥 = ∅ → ((𝜏𝜑) ↔ (𝜏𝜓)))
52, 4elab 3615 . . . . 5 (∅ ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜓))
61, 5mpbir 234 . . . 4 ∅ ∈ {𝑥 ∣ (𝜏𝜑)}
7 finds2.5 . . . . . . 7 (𝑦 ∈ ω → (𝜏 → (𝜒𝜃)))
87a2d 29 . . . . . 6 (𝑦 ∈ ω → ((𝜏𝜒) → (𝜏𝜃)))
9 vex 3444 . . . . . . 7 𝑦 ∈ V
10 finds2.2 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜒))
1110imbi2d 344 . . . . . . 7 (𝑥 = 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜒)))
129, 11elab 3615 . . . . . 6 (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜒))
139sucex 7509 . . . . . . 7 suc 𝑦 ∈ V
14 finds2.3 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝜑𝜃))
1514imbi2d 344 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜃)))
1613, 15elab 3615 . . . . . 6 (suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜃))
178, 12, 163imtr4g 299 . . . . 5 (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)}))
1817rgen 3116 . . . 4 𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)})
19 peano5 7588 . . . 4 ((∅ ∈ {𝑥 ∣ (𝜏𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏𝜑)})
206, 18, 19mp2an 691 . . 3 ω ⊆ {𝑥 ∣ (𝜏𝜑)}
2120sseli 3911 . 2 (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏𝜑)})
22 abid 2780 . 2 (𝑥 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜑))
2321, 22sylib 221 1 (𝑥 ∈ ω → (𝜏𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106   ⊆ wss 3881  ∅c0 4243  suc csuc 6162  ωcom 7563 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-tr 5138  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-om 7564 This theorem is referenced by:  finds1  7595  onnseq  7967  nnacl  8223  nnmcl  8224  nnecl  8225  nnacom  8229  nnaass  8234  nndi  8235  nnmass  8236  nnmsucr  8237  nnmcom  8238  nnmordi  8243  omsmolem  8266  isinf  8718  unblem2  8758  fiint  8782  dffi3  8882  card2inf  9006  cantnfle  9121  cantnflt  9122  cantnflem1  9139  cnfcom  9150  trcl  9157  fseqenlem1  9438  nnadju  9611  infpssrlem3  9719  fin23lem26  9739  axdc3lem2  9865  axdc4lem  9869  axdclem2  9934  wunr1om  10133  wuncval2  10161  tskr1om  10181  grothomex  10243  peano5nni  11631  neibastop2lem  33836  finxpreclem6  34832  domalom  34840
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