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Theorem arch 11249
 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 variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 breq1 4626 . . 3 (𝑦 = 𝐴 → (𝑦 < 𝑛𝐴 < 𝑛))
21rexbidv 3047 . 2 (𝑦 = 𝐴 → (∃𝑛 ∈ ℕ 𝑦 < 𝑛 ↔ ∃𝑛 ∈ ℕ 𝐴 < 𝑛))
3 nnunb 11248 . . . 4 ¬ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦)
4 ralnex 2988 . . . 4 (∀𝑦 ∈ ℝ ¬ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦) ↔ ¬ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦))
53, 4mpbir 221 . . 3 𝑦 ∈ ℝ ¬ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦)
6 rexnal 2991 . . . . 5 (∃𝑛 ∈ ℕ ¬ (𝑛 < 𝑦𝑛 = 𝑦) ↔ ¬ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦))
7 nnre 10987 . . . . . . . . 9 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
8 axlttri 10069 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑦 < 𝑛 ↔ ¬ (𝑦 = 𝑛𝑛 < 𝑦)))
97, 8sylan2 491 . . . . . . . 8 ((𝑦 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑦 < 𝑛 ↔ ¬ (𝑦 = 𝑛𝑛 < 𝑦)))
10 equcom 1942 . . . . . . . . . . 11 (𝑦 = 𝑛𝑛 = 𝑦)
1110orbi1i 542 . . . . . . . . . 10 ((𝑦 = 𝑛𝑛 < 𝑦) ↔ (𝑛 = 𝑦𝑛 < 𝑦))
12 orcom 402 . . . . . . . . . 10 ((𝑛 = 𝑦𝑛 < 𝑦) ↔ (𝑛 < 𝑦𝑛 = 𝑦))
1311, 12bitri 264 . . . . . . . . 9 ((𝑦 = 𝑛𝑛 < 𝑦) ↔ (𝑛 < 𝑦𝑛 = 𝑦))
1413notbii 310 . . . . . . . 8 (¬ (𝑦 = 𝑛𝑛 < 𝑦) ↔ ¬ (𝑛 < 𝑦𝑛 = 𝑦))
159, 14syl6bb 276 . . . . . . 7 ((𝑦 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑦 < 𝑛 ↔ ¬ (𝑛 < 𝑦𝑛 = 𝑦)))
1615biimprd 238 . . . . . 6 ((𝑦 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (¬ (𝑛 < 𝑦𝑛 = 𝑦) → 𝑦 < 𝑛))
1716reximdva 3013 . . . . 5 (𝑦 ∈ ℝ → (∃𝑛 ∈ ℕ ¬ (𝑛 < 𝑦𝑛 = 𝑦) → ∃𝑛 ∈ ℕ 𝑦 < 𝑛))
186, 17syl5bir 233 . . . 4 (𝑦 ∈ ℝ → (¬ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦) → ∃𝑛 ∈ ℕ 𝑦 < 𝑛))
1918ralimia 2946 . . 3 (∀𝑦 ∈ ℝ ¬ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦) → ∀𝑦 ∈ ℝ ∃𝑛 ∈ ℕ 𝑦 < 𝑛)
205, 19ax-mp 5 . 2 𝑦 ∈ ℝ ∃𝑛 ∈ ℕ 𝑦 < 𝑛
212, 20vtoclri 3273 1 (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2908  ∃wrex 2909   class class class wbr 4623  ℝcr 9895   < clt 10034  ℕcn 10980 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981 This theorem is referenced by:  nnrecl  11250  bndndx  11251  btwnz  11439  uzwo3  11743  zmin  11744  rpnnen1lem5  11778  rpnnen1lem5OLD  11784  harmonic  14535  alzdvds  14985  ovolicc2lem4  23228  volsup2  23313  ismbf3d  23361  mbfi1fseqlem6  23427  itg2seq  23449  itg2cnlem1  23468  ply1divex  23834  plydivex  23990  lgamucov  24698  lgamcvg2  24715  ubthlem1  27614  lnconi  28780  rearchi  29669  esumcst  29948  hbtlem5  37218  prmunb2  38031  rfcnnnub  38717  stoweidlem14  39568  stoweidlem60  39614  sge0rpcpnf  39975  hoicvr  40099
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