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Theorem nnind 8111
Description: Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 8115 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
Hypotheses
Ref Expression
nnind.1 (𝑥 = 1 → (𝜑𝜓))
nnind.2 (𝑥 = 𝑦 → (𝜑𝜒))
nnind.3 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
nnind.4 (𝑥 = 𝐴 → (𝜑𝜏))
nnind.5 𝜓
nnind.6 (𝑦 ∈ ℕ → (𝜒𝜃))
Assertion
Ref Expression
nnind (𝐴 ∈ ℕ → 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 nnind
StepHypRef Expression
1 1nn 8106 . . . . . 6 1 ∈ ℕ
2 nnind.5 . . . . . 6 𝜓
3 nnind.1 . . . . . . 7 (𝑥 = 1 → (𝜑𝜓))
43elrab 2750 . . . . . 6 (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓))
51, 2, 4mpbir2an 884 . . . . 5 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑}
6 elrabi 2747 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ)
7 peano2nn 8107 . . . . . . . . . 10 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
87a1d 22 . . . . . . . . 9 (𝑦 ∈ ℕ → (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ))
9 nnind.6 . . . . . . . . 9 (𝑦 ∈ ℕ → (𝜒𝜃))
108, 9anim12d 328 . . . . . . . 8 (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃)))
11 nnind.2 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜒))
1211elrab 2750 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒))
13 nnind.3 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
1413elrab 2750 . . . . . . . 8 ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃))
1510, 12, 143imtr4g 203 . . . . . . 7 (𝑦 ∈ ℕ → (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
166, 15mpcom 36 . . . . . 6 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
1716rgen 2417 . . . . 5 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
18 peano5nni 8098 . . . . 5 ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑})
195, 17, 18mp2an 417 . . . 4 ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}
2019sseli 2996 . . 3 (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑})
21 nnind.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜏))
2221elrab 2750 . . 3 (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏))
2320, 22sylib 120 . 2 (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏))
2423simprd 112 1 (𝐴 ∈ ℕ → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  wral 2349  {crab 2353  wss 2974  (class class class)co 5537  1c1 7033   + caddc 7035  cn 8095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-cnex 7118  ax-resscn 7119  ax-1re 7121  ax-addrcl 7124
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-br 3788  df-iota 4891  df-fv 4934  df-ov 5540  df-inn 8096
This theorem is referenced by:  nnindALT  8112  nn1m1nn  8113  nnaddcl  8115  nnmulcl  8116  nnge1  8118  nn1gt1  8128  nnsub  8133  zaddcllempos  8458  zaddcllemneg  8460  nneoor  8519  peano5uzti  8525  nn0ind-raph  8534  indstr  8751  exbtwnzlemshrink  9324  expivallem  9563  expcllem  9573  expap0  9592  resqrexlemover  10023  resqrexlemlo  10026  resqrexlemcalc3  10029  gcdmultiple  10542  rplpwr  10549  prmind2  10635  prmdvdsexp  10660  sqrt2irr  10674  pw2dvdslemn  10676
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