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Theorem finds 4484
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 4025 . . . . . 6 ∅ ∈ V
3 finds.1 . . . . . 6 (𝑥 = ∅ → (𝜑𝜓))
42, 3elab 2802 . . . . 5 (∅ ∈ {𝑥𝜑} ↔ 𝜓)
51, 4mpbir 145 . . . 4 ∅ ∈ {𝑥𝜑}
6 finds.6 . . . . . 6 (𝑦 ∈ ω → (𝜒𝜃))
7 vex 2663 . . . . . . 7 𝑦 ∈ V
8 finds.2 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜒))
97, 8elab 2802 . . . . . 6 (𝑦 ∈ {𝑥𝜑} ↔ 𝜒)
107sucex 4385 . . . . . . 7 suc 𝑦 ∈ V
11 finds.3 . . . . . . 7 (𝑥 = suc 𝑦 → (𝜑𝜃))
1210, 11elab 2802 . . . . . 6 (suc 𝑦 ∈ {𝑥𝜑} ↔ 𝜃)
136, 9, 123imtr4g 204 . . . . 5 (𝑦 ∈ ω → (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑}))
1413rgen 2462 . . . 4 𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})
15 peano5 4482 . . . 4 ((∅ ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})) → ω ⊆ {𝑥𝜑})
165, 14, 15mp2an 422 . . 3 ω ⊆ {𝑥𝜑}
1716sseli 3063 . 2 (𝐴 ∈ ω → 𝐴 ∈ {𝑥𝜑})
18 finds.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
1918elabg 2803 . 2 (𝐴 ∈ ω → (𝐴 ∈ {𝑥𝜑} ↔ 𝜏))
2017, 19mpbid 146 1 (𝐴 ∈ ω → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wcel 1465  {cab 2103  wral 2393  wss 3041  c0 3333  suc csuc 4257  ωcom 4474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-suc 4263  df-iom 4475
This theorem is referenced by:  findes  4487  nn0suc  4488  elnn  4489  ordom  4490  nndceq0  4501  0elnn  4502  omsinds  4505  nna0r  6342  nnm0r  6343  nnsucelsuc  6355  nneneq  6719  php5  6720  php5dom  6725  fidcenumlemrk  6810  fidcenumlemr  6811  frec2uzltd  10144  frecuzrdgg  10157  seq3val  10199  seqvalcd  10200  omgadd  10516  zfz1iso  10552  ennnfonelemhom  11855  nninfalllemn  13129  nninfsellemdc  13133
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