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Theorem arch 9171
Description: Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
Assertion
Ref Expression
arch (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
Distinct variable group:   𝐴,𝑛

Proof of Theorem arch
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-arch 7929 . . 3 (𝐴 ∈ ℝ → ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
2 dfnn2 8919 . . . 4 ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
32rexeqi 2677 . . 3 (∃𝑛 ∈ ℕ 𝐴 < 𝑛 ↔ ∃𝑛 {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 < 𝑛)
41, 3sylibr 134 . 2 (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
5 nnre 8924 . . . 4 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
6 ltxrlt 8021 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝐴 < 𝑛𝐴 < 𝑛))
75, 6sylan2 286 . . 3 ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝐴 < 𝑛𝐴 < 𝑛))
87rexbidva 2474 . 2 (𝐴 ∈ ℝ → (∃𝑛 ∈ ℕ 𝐴 < 𝑛 ↔ ∃𝑛 ∈ ℕ 𝐴 < 𝑛))
94, 8mpbird 167 1 (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wcel 2148  {cab 2163  wral 2455  wrex 2456   cint 3844   class class class wbr 4003  (class class class)co 5874  cr 7809  1c1 7811   + caddc 7813   < cltrr 7814   < clt 7990  cn 8917
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-in1 614  ax-in2 615  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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907  ax-arch 7929
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-xp 4632  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918
This theorem is referenced by:  nnrecl  9172  bndndx  9173  btwnz  9370  expnbnd  10640  cvg1nlemres  10989  cvg1n  10990  resqrexlemga  11027  fsum3cvg3  11399  divcnv  11500  efcllem  11662  alzdvds  11854  dvdsbnd  11951
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