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Theorem finds2 6963
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 4713 . . . . . 6 ∅ ∈ V
3 finds2.1 . . . . . . 7 (𝑥 = ∅ → (𝜑𝜓))
43imbi2d 328 . . . . . 6 (𝑥 = ∅ → ((𝜏𝜑) ↔ (𝜏𝜓)))
52, 4elab 3318 . . . . 5 (∅ ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜓))
61, 5mpbir 219 . . . 4 ∅ ∈ {𝑥 ∣ (𝜏𝜑)}
7 finds2.5 . . . . . . 7 (𝑦 ∈ ω → (𝜏 → (𝜒𝜃)))
87a2d 29 . . . . . 6 (𝑦 ∈ ω → ((𝜏𝜒) → (𝜏𝜃)))
9 vex 3175 . . . . . . 7 𝑦 ∈ V
10 finds2.2 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜒))
1110imbi2d 328 . . . . . . 7 (𝑥 = 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜒)))
129, 11elab 3318 . . . . . 6 (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜒))
139sucex 6880 . . . . . . 7 suc 𝑦 ∈ V
14 finds2.3 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝜑𝜃))
1514imbi2d 328 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜃)))
1613, 15elab 3318 . . . . . 6 (suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜃))
178, 12, 163imtr4g 283 . . . . 5 (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)}))
1817rgen 2905 . . . 4 𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)})
19 peano5 6958 . . . 4 ((∅ ∈ {𝑥 ∣ (𝜏𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏𝜑)})
206, 18, 19mp2an 703 . . 3 ω ⊆ {𝑥 ∣ (𝜏𝜑)}
2120sseli 3563 . 2 (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏𝜑)})
22 abid 2597 . 2 (𝑥 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜑))
2321, 22sylib 206 1 (𝑥 ∈ ω → (𝜏𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976  {cab 2595  wral 2895  wss 3539  c0 3873  suc csuc 5628  ωcom 6934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-om 6935
This theorem is referenced by:  finds1  6964  onnseq  7305  nnacl  7555  nnmcl  7556  nnecl  7557  nnacom  7561  nnaass  7566  nndi  7567  nnmass  7568  nnmsucr  7569  nnmcom  7570  nnmordi  7575  omsmolem  7597  isinf  8035  unblem2  8075  fiint  8099  dffi3  8197  card2inf  8320  cantnfle  8428  cantnflt  8429  cantnflem1  8446  cnfcom  8457  trcl  8464  fseqenlem1  8707  infpssrlem3  8987  fin23lem26  9007  axdc3lem2  9133  axdc4lem  9137  axdclem2  9202  wunr1om  9397  wuncval2  9425  tskr1om  9445  grothomex  9507  peano5nni  10870  neibastop2lem  31331  finxpreclem6  32205
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