Theorem nnind 9322

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 9326 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 → (𝜑 ↔ 𝜓))
nnind.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
nnind.3	⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
nnind.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
nnind.5	⊢ 𝜓
nnind.6	⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))

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

Step	Hyp	Ref	Expression
1		1nn 9317	. . . . . 6 ⊢ 1 ∈ ℕ
2		nnind.5	. . . . . 6 ⊢ 𝜓
3		nnind.1	. . . . . . 7 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))
4	3	elrab 2982	. . . . . 6 ⊢ (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓))
5	1, 2, 4	mpbir2an 955	. . . . 5 ⊢ 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑}
6		elrabi 2979	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ)
7		peano2nn 9318	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
8	7	a1d 22	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ))
9		nnind.6	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))
10	8, 9	anim12d 335	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃)))
11		nnind.2	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
12	11	elrab 2982	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒))
13		nnind.3	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
14	13	elrab 2982	. . . . . . . 8 ⊢ ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃))
15	10, 12, 14	3imtr4g 205	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
16	6, 15	mpcom 36	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
17	16	rgen 2603	. . . . 5 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
18		peano5nni 9309	. . . . 5 ⊢ ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑})
19	5, 17, 18	mp2an 430	. . . 4 ⊢ ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}
20	19	sseli 3244	. . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑})
21		nnind.4	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
22	21	elrab 2982	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏))
23	20, 22	sylib 122	. 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏))
24	23	simprd 114	1 ⊢ (𝐴 ∈ ℕ → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532   ⊆ wss 3220  (class class class)co 6085  1c1 8180   + caddc 8182  ℕcn 9306

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4249  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088  df-inn 9307

This theorem is used by:  nnindALT  9323  nn1m1nn  9324  nnaddcl  9326  nnmulcl  9327  nnge1  9329  nn1gt1  9340  nnsub  9345  zaddcllempos  9685  zaddcllemneg  9687  nneoor  9752  peano5uzti  9758  nn0ind-raph  9767  indstr  10002  exbtwnzlemshrink  10693  exp3vallem  10990  expcllem  11000  expap0  11019  apexp1  11170  seq3coll  11308  resqrexlemover  11790  resqrexlemlo  11793  resqrexlemcalc3  11796  gcdmultiple  12813  rplpwr  12820  prmind2  12914  prmdvdsexp  12943  sqrt2irr  12957  pwbdvdslemn  12960  pcmpt  13142  prmpwdvds  13154  mulgnnass  14009  dvexp  15861  plycolemc  15908  ppiqltx  16183  bposlem5  16213  2sqlem10  16342