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Mirrors > Home > ILE Home > Th. List > btwnz | GIF version |
Description: Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.) |
Ref | Expression |
---|---|
btwnz | ⊢ (𝐴 ∈ ℝ → (∃𝑥 ∈ ℤ 𝑥 < 𝐴 ∧ ∃𝑦 ∈ ℤ 𝐴 < 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcl 8208 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
2 | arch 9162 | . . . 4 ⊢ (-𝐴 ∈ ℝ → ∃𝑧 ∈ ℕ -𝐴 < 𝑧) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃𝑧 ∈ ℕ -𝐴 < 𝑧) |
4 | nnre 8915 | . . . . . . . 8 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℝ) | |
5 | ltnegcon1 8410 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-𝐴 < 𝑧 ↔ -𝑧 < 𝐴)) | |
6 | 5 | ex 115 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝑧 ∈ ℝ → (-𝐴 < 𝑧 ↔ -𝑧 < 𝐴))) |
7 | 4, 6 | syl5 32 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝑧 ∈ ℕ → (-𝐴 < 𝑧 ↔ -𝑧 < 𝐴))) |
8 | 7 | pm5.32d 450 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑧 ∈ ℕ ∧ -𝐴 < 𝑧) ↔ (𝑧 ∈ ℕ ∧ -𝑧 < 𝐴))) |
9 | nnnegz 9245 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → -𝑧 ∈ ℤ) | |
10 | breq1 4003 | . . . . . . . 8 ⊢ (𝑥 = -𝑧 → (𝑥 < 𝐴 ↔ -𝑧 < 𝐴)) | |
11 | 10 | rspcev 2841 | . . . . . . 7 ⊢ ((-𝑧 ∈ ℤ ∧ -𝑧 < 𝐴) → ∃𝑥 ∈ ℤ 𝑥 < 𝐴) |
12 | 9, 11 | sylan 283 | . . . . . 6 ⊢ ((𝑧 ∈ ℕ ∧ -𝑧 < 𝐴) → ∃𝑥 ∈ ℤ 𝑥 < 𝐴) |
13 | 8, 12 | syl6bi 163 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝑧 ∈ ℕ ∧ -𝐴 < 𝑧) → ∃𝑥 ∈ ℤ 𝑥 < 𝐴)) |
14 | 13 | expd 258 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝑧 ∈ ℕ → (-𝐴 < 𝑧 → ∃𝑥 ∈ ℤ 𝑥 < 𝐴))) |
15 | 14 | rexlimdv 2593 | . . 3 ⊢ (𝐴 ∈ ℝ → (∃𝑧 ∈ ℕ -𝐴 < 𝑧 → ∃𝑥 ∈ ℤ 𝑥 < 𝐴)) |
16 | 3, 15 | mpd 13 | . 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ 𝑥 < 𝐴) |
17 | arch 9162 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃𝑦 ∈ ℕ 𝐴 < 𝑦) | |
18 | nnz 9261 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ) | |
19 | 18 | anim1i 340 | . . . 4 ⊢ ((𝑦 ∈ ℕ ∧ 𝐴 < 𝑦) → (𝑦 ∈ ℤ ∧ 𝐴 < 𝑦)) |
20 | 19 | reximi2 2573 | . . 3 ⊢ (∃𝑦 ∈ ℕ 𝐴 < 𝑦 → ∃𝑦 ∈ ℤ 𝐴 < 𝑦) |
21 | 17, 20 | syl 14 | . 2 ⊢ (𝐴 ∈ ℝ → ∃𝑦 ∈ ℤ 𝐴 < 𝑦) |
22 | 16, 21 | jca 306 | 1 ⊢ (𝐴 ∈ ℝ → (∃𝑥 ∈ ℤ 𝑥 < 𝐴 ∧ ∃𝑦 ∈ ℤ 𝐴 < 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2148 ∃wrex 2456 class class class wbr 4000 ℝcr 7801 < clt 7982 -cneg 8119 ℕcn 8908 ℤcz 9242 |
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 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-ltadd 7918 ax-arch 7921 |
This theorem depends on definitions: df-bi 117 df-3or 979 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-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-inn 8909 df-z 9243 |
This theorem is referenced by: lbzbi 9605 exbtwnzlemex 10236 rebtwn2z 10241 |
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