Theorem nnind 9299

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 9303 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 9294	. . . . . 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 9295	. . . . . . . . . 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 9286	. . . . 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)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532   ⊆ wss 3220  (class class class)co 6075  1c1 8170   + caddc 8172  ℕcn 9283

This theorem was proved from 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 4244  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem 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 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078  df-inn 9284

This theorem is referenced by:  nnindALT  9300  nn1m1nn  9301  nnaddcl  9303  nnmulcl  9304  nnge1  9306  nn1gt1  9317  nnsub  9322  zaddcllempos  9660  zaddcllemneg  9662  nneoor  9727  peano5uzti  9733  nn0ind-raph  9742  indstr  9972  exbtwnzlemshrink  10661  exp3vallem  10955  expcllem  10965  expap0  10984  apexp1  11134  seq3coll  11272  resqrexlemover  11754  resqrexlemlo  11757  resqrexlemcalc3  11760  gcdmultiple  12775  rplpwr  12782  prmind2  12876  prmdvdsexp  12904  sqrt2irr  12918  pw2dvdslemn  12921  pcmpt  13100  prmpwdvds  13112  mulgnnass  13937  dvexp  15735  plycolemc  15782  2sqlem10  16158