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Theorem arch 10957
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 4437 . . 3 (𝑦 = 𝐴 → (𝑦 < 𝑛𝐴 < 𝑛))
21rexbidv 2927 . 2 (𝑦 = 𝐴 → (∃𝑛 ∈ ℕ 𝑦 < 𝑛 ↔ ∃𝑛 ∈ ℕ 𝐴 < 𝑛))
3 nnunb 10956 . . . 4 ¬ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦)
4 ralnex 2869 . . . 4 (∀𝑦 ∈ ℝ ¬ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦) ↔ ¬ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦))
53, 4mpbir 216 . . 3 𝑦 ∈ ℝ ¬ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦)
6 rexnal 2871 . . . . 5 (∃𝑛 ∈ ℕ ¬ (𝑛 < 𝑦𝑛 = 𝑦) ↔ ¬ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦))
7 nnre 10705 . . . . . . . . 9 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
8 axlttri 9790 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑦 < 𝑛 ↔ ¬ (𝑦 = 𝑛𝑛 < 𝑦)))
97, 8sylan2 484 . . . . . . . 8 ((𝑦 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑦 < 𝑛 ↔ ¬ (𝑦 = 𝑛𝑛 < 𝑦)))
10 equcom 1894 . . . . . . . . . . 11 (𝑦 = 𝑛𝑛 = 𝑦)
1110orbi1i 535 . . . . . . . . . 10 ((𝑦 = 𝑛𝑛 < 𝑦) ↔ (𝑛 = 𝑦𝑛 < 𝑦))
12 orcom 396 . . . . . . . . . 10 ((𝑛 = 𝑦𝑛 < 𝑦) ↔ (𝑛 < 𝑦𝑛 = 𝑦))
1311, 12bitri 259 . . . . . . . . 9 ((𝑦 = 𝑛𝑛 < 𝑦) ↔ (𝑛 < 𝑦𝑛 = 𝑦))
1413notbii 305 . . . . . . . 8 (¬ (𝑦 = 𝑛𝑛 < 𝑦) ↔ ¬ (𝑛 < 𝑦𝑛 = 𝑦))
159, 14syl6bb 271 . . . . . . 7 ((𝑦 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑦 < 𝑛 ↔ ¬ (𝑛 < 𝑦𝑛 = 𝑦)))
1615biimprd 233 . . . . . 6 ((𝑦 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (¬ (𝑛 < 𝑦𝑛 = 𝑦) → 𝑦 < 𝑛))
1716reximdva 2893 . . . . 5 (𝑦 ∈ ℝ → (∃𝑛 ∈ ℕ ¬ (𝑛 < 𝑦𝑛 = 𝑦) → ∃𝑛 ∈ ℕ 𝑦 < 𝑛))
186, 17syl5bir 228 . . . 4 (𝑦 ∈ ℝ → (¬ ∀𝑛 ∈ ℕ (𝑛 < 𝑦𝑛 = 𝑦) → ∃𝑛 ∈ ℕ 𝑦 < 𝑛))
1918ralimia 2829 . . 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 3145 1 (𝐴 ∈ ℝ → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 191  wo 377  wa 378   = wceq 1468  wcel 1937  wral 2791  wrex 2792   class class class wbr 4434  cr 9623   < clt 9760  cn 10698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659  ax-resscn 9681  ax-1cn 9682  ax-icn 9683  ax-addcl 9684  ax-addrcl 9685  ax-mulcl 9686  ax-mulrcl 9687  ax-mulcom 9688  ax-addass 9689  ax-mulass 9690  ax-distr 9691  ax-i2m1 9692  ax-1ne0 9693  ax-1rid 9694  ax-rnegex 9695  ax-rrecex 9696  ax-cnre 9697  ax-pre-lttri 9698  ax-pre-lttrn 9699  ax-pre-ltadd 9700  ax-pre-mulgt0 9701  ax-pre-sup 9702
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-nel 2678  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 5431  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-riota 6325  df-ov 6366  df-oprab 6367  df-mpt2 6368  df-om 6770  df-wrecs 7105  df-recs 7167  df-rdg 7205  df-er 7440  df-en 7653  df-dom 7654  df-sdom 7655  df-pnf 9762  df-mnf 9763  df-xr 9764  df-ltxr 9765  df-le 9766  df-sub 9949  df-neg 9950  df-nn 10699
This theorem is referenced by:  nnrecl  10958  bndndx  10959  btwnz  11127  uzwo3  11350  zmin  11351  rpnnen1lem5  11385  harmonic  14077  alzdvds  14515  ovolicc2lem4OLD  22632  ovolicc2lem4  22633  volsup2  22723  ismbf3d  22771  mbfi1fseqlem6  22839  itg2seq  22861  itg2cnlem1  22880  ply1divex  23248  plydivex  23411  lgamucov  24124  lgamcvg2  24141  ubthlem1  26675  lnconi  27849  rearchi  28760  esumcst  29039  hbtlem5  36227  prmunb2  37016  rfcnnnub  37705  stoweidlem14  38310  stoweidlem60  38357  sge0rpcpnf  38709  hoicvr  38833
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