ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  peano2nn GIF version

Theorem peano2nn 8931
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.
StepHypRef Expression
1 dfnn2 8921 . . . . . 6 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
21eleq2i 2244 . . . . 5 (𝐴 ∈ ℕ ↔ 𝐴 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
3 elintg 3853 . . . . 5 (𝐴 ∈ ℕ → (𝐴 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴𝑧))
42, 3bitrid 192 . . . 4 (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴𝑧))
54ibi 176 . . 3 (𝐴 ∈ ℕ → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴𝑧)
6 vex 2741 . . . . . . . 8 𝑧 ∈ V
7 eleq2 2241 . . . . . . . . 9 (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
8 eleq2 2241 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
98raleqbi1dv 2681 . . . . . . . . 9 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧))
107, 9anbi12d 473 . . . . . . . 8 (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)))
116, 10elab 2882 . . . . . . 7 (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧))
1211simprbi 275 . . . . . 6 (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} → ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)
13 oveq1 5882 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦 + 1) = (𝐴 + 1))
1413eleq1d 2246 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦 + 1) ∈ 𝑧 ↔ (𝐴 + 1) ∈ 𝑧))
1514rspcva 2840 . . . . . 6 ((𝐴𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧) → (𝐴 + 1) ∈ 𝑧)
1612, 15sylan2 286 . . . . 5 ((𝐴𝑧𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}) → (𝐴 + 1) ∈ 𝑧)
1716expcom 116 . . . 4 (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} → (𝐴𝑧 → (𝐴 + 1) ∈ 𝑧))
1817ralimia 2538 . . 3 (∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴𝑧 → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧)
195, 18syl 14 . 2 (𝐴 ∈ ℕ → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧)
20 nnre 8926 . . . 4 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
21 1red 7972 . . . 4 (𝐴 ∈ ℕ → 1 ∈ ℝ)
2220, 21readdcld 7987 . . 3 (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℝ)
231eleq2i 2244 . . . 4 ((𝐴 + 1) ∈ ℕ ↔ (𝐴 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
24 elintg 3853 . . . 4 ((𝐴 + 1) ∈ ℝ → ((𝐴 + 1) ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧))
2523, 24bitrid 192 . . 3 ((𝐴 + 1) ∈ ℝ → ((𝐴 + 1) ∈ ℕ ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧))
2622, 25syl 14 . 2 (𝐴 ∈ ℕ → ((𝐴 + 1) ∈ ℕ ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} (𝐴 + 1) ∈ 𝑧))
2719, 26mpbird 167 1 (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  {cab 2163  wral 2455   cint 3845  (class class class)co 5875  cr 7810  1c1 7812   + caddc 7814  cn 8919
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4122  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-iota 5179  df-fv 5225  df-ov 5878  df-inn 8920
This theorem is referenced by:  peano2nnd  8934  nnind  8935  nnaddcl  8939  2nn  9080  3nn  9081  4nn  9082  5nn  9083  6nn  9084  7nn  9085  8nn  9086  9nn  9087  nneoor  9355  10nn  9399  nnsplit  10137  fzonn0p1p1  10213  expp1  10527  facp1  10710  resqrexlemfp1  11018  resqrexlemcalc3  11025  trireciplem  11508  trirecip  11509  cvgratnnlemnexp  11532  cvgratz  11540  nno  11911  nnoddm1d2  11915  rplpwr  12028  prmind2  12120  sqrt2irr  12162  pcmpt  12341  pockthi  12356  mulgnnp1  12991  2sqlem10  14475
  Copyright terms: Public domain W3C validator