Theorem peano2nn 9316

Description: Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
peano2nn	⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)

Proof of Theorem peano2nn

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfnn2 9306	. . . . . 6 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
2	1	eleq2i 2305	. . . . 5 ⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
3		elintg 3978	. . . . 5 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 ∈ 𝑧))
4	2, 3	bitrid 192	. . . 4 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 ∈ 𝑧))
5	4	ibi 176	. . 3 ⊢ (𝐴 ∈ ℕ → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 ∈ 𝑧)
6		vex 2824	. . . . . . . 8 ⊢ 𝑧 ∈ V
7		eleq2 2302	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
8		eleq2 2302	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
9	8	raleqbi1dv 2761	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))
10	7, 9	anbi12d 477	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)))
11	6, 10	elab 2970	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))
12	11	simprbi 275	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)
13		oveq1 6092	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 + 1) = (𝐴 + 1))
14	13	eleq1d 2307	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝑦 + 1) ∈ 𝑧 ↔ (𝐴 + 1) ∈ 𝑧))
15	14	rspcva 2927	. . . . . 6 ⊢ ((𝐴 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → (𝐴 + 1) ∈ 𝑧)
16	12, 15	sylan2 286	. . . . 5 ⊢ ((𝐴 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}) → (𝐴 + 1) ∈ 𝑧)
17	16	expcom 116	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → (𝐴 ∈ 𝑧 → (𝐴 + 1) ∈ 𝑧))
18	17	ralimia 2611	. . 3 ⊢ (∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 ∈ 𝑧 → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧)
19	5, 18	syl 14	. 2 ⊢ (𝐴 ∈ ℕ → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧)
20		nnre 9311	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
21		1red 8341	. . . 4 ⊢ (𝐴 ∈ ℕ → 1 ∈ ℝ)
22	20, 21	readdcld 8355	. . 3 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℝ)
23	1	eleq2i 2305	. . . 4 ⊢ ((𝐴 + 1) ∈ ℕ ↔ (𝐴 + 1) ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
24		elintg 3978	. . . 4 ⊢ ((𝐴 + 1) ∈ ℝ → ((𝐴 + 1) ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧))
25	23, 24	bitrid 192	. . 3 ⊢ ((𝐴 + 1) ∈ ℝ → ((𝐴 + 1) ∈ ℕ ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧))
26	22, 25	syl 14	. 2 ⊢ (𝐴 ∈ ℕ → ((𝐴 + 1) ∈ ℕ ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧))
27	19, 26	mpbird 167	1 ⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∩ cint 3970  (class class class)co 6085  ℝcr 8178  1c1 8180   + caddc 8182  ℕcn 9304

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-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 9305

This theorem is used by:  peano2nnd  9319  nnind  9320  nnaddcl  9324  2nn  9466  3nn  9467  4nn  9468  5nn  9469  6nn  9470  7nn  9471  8nn  9472  9nn  9473  nneoor  9748  10nn  9792  nnsplit  10544  fzonn0p1p1  10631  expp1  10983  facp1  11168  resqrexlemfp1  11775  resqrexlemcalc3  11782  trireciplem  12267  trirecip  12268  cvgratnnlemnexp  12291  cvgratz  12299  nno  12673  nnoddm1d2  12677  rplpwr  12804  prmind2  12898  sqrt2irr  12940  pcmpt  13122  pockthi  13137  dec5nprm  13193  mulgnnp1  13933  2sqlem10  16244