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Theorem finds 7237
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 4924 . . . . . 6 ∅ ∈ V
3 finds.1 . . . . . 6 (𝑥 = ∅ → (𝜑𝜓))
42, 3elab 3501 . . . . 5 (∅ ∈ {𝑥𝜑} ↔ 𝜓)
51, 4mpbir 221 . . . 4 ∅ ∈ {𝑥𝜑}
6 finds.6 . . . . . 6 (𝑦 ∈ ω → (𝜒𝜃))
7 vex 3354 . . . . . . 7 𝑦 ∈ V
8 finds.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
97, 8elab 3501 . . . . . 6 (𝑦 ∈ {𝑥𝜑} ↔ 𝜒)
107sucex 7156 . . . . . . 7 suc 𝑦 ∈ V
11 finds.3 . . . . . . 7 (𝑥 = suc 𝑦 → (𝜑𝜃))
1210, 11elab 3501 . . . . . 6 (suc 𝑦 ∈ {𝑥𝜑} ↔ 𝜃)
136, 9, 123imtr4g 285 . . . . 5 (𝑦 ∈ ω → (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑}))
1413rgen 3071 . . . 4 𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})
15 peano5 7234 . . . 4 ((∅ ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})) → ω ⊆ {𝑥𝜑})
165, 14, 15mp2an 672 . . 3 ω ⊆ {𝑥𝜑}
1716sseli 3748 . 2 (𝐴 ∈ ω → 𝐴 ∈ {𝑥𝜑})
18 finds.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
1918elabg 3502 . 2 (𝐴 ∈ ω → (𝐴 ∈ {𝑥𝜑} ↔ 𝜏))
2017, 19mpbid 222 1 (𝐴 ∈ ω → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wss 3723  c0 4063  suc csuc 5866  ωcom 7210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-om 7211
This theorem is referenced by:  findsg  7238  findes  7241  seqomlem1  7696  nna0r  7841  nnm0r  7842  nnawordi  7853  nneob  7884  nneneq  8297  pssnn  8332  inf3lem1  8687  inf3lem2  8688  cantnfval2  8728  cantnfp1lem3  8739  r1fin  8798  ackbij1lem14  9255  ackbij1lem16  9257  ackbij1  9260  ackbij2lem2  9262  ackbij2lem3  9263  infpssrlem4  9328  fin23lem14  9355  fin23lem34  9368  itunitc1  9442  ituniiun  9444  om2uzuzi  12949  om2uzlti  12950  om2uzrdg  12956  uzrdgxfr  12967  hashgadd  13361  mreexexd  16509  trpredmintr  32060  findfvcl  32781  finxp00  33569
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