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Theorem 1nn 8117
Description: Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
Assertion
Ref Expression
1nn 1 ∈ ℕ

Proof of Theorem 1nn
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfnn2 8108 . . . 4 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
21eleq2i 2146 . . 3 (1 ∈ ℕ ↔ 1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
3 1re 7180 . . . 4 1 ∈ ℝ
4 elintg 3652 . . . 4 (1 ∈ ℝ → (1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧))
53, 4ax-mp 7 . . 3 (1 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧)
62, 5bitri 182 . 2 (1 ∈ ℕ ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}1 ∈ 𝑧)
7 vex 2605 . . . 4 𝑧 ∈ V
8 eleq2 2143 . . . . 5 (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
9 eleq2 2143 . . . . . 6 (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
109raleqbi1dv 2558 . . . . 5 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧))
118, 10anbi12d 457 . . . 4 (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧)))
127, 11elab 2739 . . 3 (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (1 ∈ 𝑧 ∧ ∀𝑦𝑧 (𝑦 + 1) ∈ 𝑧))
1312simplbi 268 . 2 (𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} → 1 ∈ 𝑧)
146, 13mprgbir 2422 1 1 ∈ ℕ
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wcel 1434  {cab 2068  wral 2349   cint 3644  (class class class)co 5543  cr 7042  1c1 7044   + caddc 7046  cn 8106
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-1re 7132
This theorem depends on definitions:  df-bi 115  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-v 2604  df-int 3645  df-inn 8107
This theorem is referenced by:  nnind  8122  nn1suc  8125  2nn  8260  1nn0  8371  nn0p1nn  8394  1z  8458  neg1z  8464  elz2  8500  nneoor  8530  9p1e10  8560  indstr  8762  elnn1uz2  8775  zq  8792  qreccl  8808  fz01or  9204  expivallem  9574  exp1  9579  nnexpcl  9586  expnbnd  9693  3dec  9739  fac1  9753  faccl  9759  faclbnd3  9767  resqrexlemf1  10032  resqrexlemcalc3  10040  resqrexlemnmsq  10041  resqrexlemnm  10042  resqrexlemcvg  10043  resqrexlemglsq  10046  resqrexlemga  10047  n2dvds1  10456  ndvdsp1  10476  gcd1  10522  bezoutr1  10566  ncoprmgcdne1b  10615  1nprm  10640  1idssfct  10641  isprm2lem  10642
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