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Mirrors > Home > MPE Home > Th. List > elfznelfzob | Structured version Visualization version GIF version |
Description: A value in a finite set of sequential integers is a border value if and only if it is not contained in the half-open integer range contained in the finite set of sequential integers. (Contributed by Alexander van der Vekens, 17-Jan-2018.) (Revised by Thierry Arnoux, 22-Dec-2021.) |
Ref | Expression |
---|---|
elfznelfzob | ⊢ (𝑀 ∈ (0...𝐾) → (¬ 𝑀 ∈ (1..^𝐾) ↔ (𝑀 = 0 ∨ 𝑀 = 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfznelfzo 13137 | . . 3 ⊢ ((𝑀 ∈ (0...𝐾) ∧ ¬ 𝑀 ∈ (1..^𝐾)) → (𝑀 = 0 ∨ 𝑀 = 𝐾)) | |
2 | 1 | ex 416 | . 2 ⊢ (𝑀 ∈ (0...𝐾) → (¬ 𝑀 ∈ (1..^𝐾) → (𝑀 = 0 ∨ 𝑀 = 𝐾))) |
3 | elfzole1 13041 | . . . . . 6 ⊢ (𝑀 ∈ (1..^𝐾) → 1 ≤ 𝑀) | |
4 | elfzolt2 13042 | . . . . . . 7 ⊢ (𝑀 ∈ (1..^𝐾) → 𝑀 < 𝐾) | |
5 | elfzoel2 13032 | . . . . . . 7 ⊢ (𝑀 ∈ (1..^𝐾) → 𝐾 ∈ ℤ) | |
6 | elfzoelz 13033 | . . . . . . 7 ⊢ (𝑀 ∈ (1..^𝐾) → 𝑀 ∈ ℤ) | |
7 | 0lt1 11151 | . . . . . . . . . . 11 ⊢ 0 < 1 | |
8 | breq1 5033 | . . . . . . . . . . 11 ⊢ (𝑀 = 0 → (𝑀 < 1 ↔ 0 < 1)) | |
9 | 7, 8 | mpbiri 261 | . . . . . . . . . 10 ⊢ (𝑀 = 0 → 𝑀 < 1) |
10 | zre 11973 | . . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
11 | 10 | adantl 485 | . . . . . . . . . . 11 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℝ) |
12 | 1red 10631 | . . . . . . . . . . 11 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → 1 ∈ ℝ) | |
13 | 11, 12 | ltnled 10776 | . . . . . . . . . 10 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → (𝑀 < 1 ↔ ¬ 1 ≤ 𝑀)) |
14 | 9, 13 | syl5ib 247 | . . . . . . . . 9 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → (𝑀 = 0 → ¬ 1 ≤ 𝑀)) |
15 | 14 | con2d 136 | . . . . . . . 8 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → (1 ≤ 𝑀 → ¬ 𝑀 = 0)) |
16 | zre 11973 | . . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
17 | ltlen 10730 | . . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (𝑀 < 𝐾 ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≠ 𝑀))) | |
18 | 10, 16, 17 | syl2anr 599 | . . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 < 𝐾 ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≠ 𝑀))) |
19 | necom 3040 | . . . . . . . . . . . . . . 15 ⊢ (𝐾 ≠ 𝑀 ↔ 𝑀 ≠ 𝐾) | |
20 | df-ne 2988 | . . . . . . . . . . . . . . 15 ⊢ (𝑀 ≠ 𝐾 ↔ ¬ 𝑀 = 𝐾) | |
21 | 19, 20 | sylbb 222 | . . . . . . . . . . . . . 14 ⊢ (𝐾 ≠ 𝑀 → ¬ 𝑀 = 𝐾) |
22 | 21 | adantl 485 | . . . . . . . . . . . . 13 ⊢ ((𝑀 ≤ 𝐾 ∧ 𝐾 ≠ 𝑀) → ¬ 𝑀 = 𝐾) |
23 | 18, 22 | syl6bi 256 | . . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 < 𝐾 → ¬ 𝑀 = 𝐾)) |
24 | 23 | ex 416 | . . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀 < 𝐾 → ¬ 𝑀 = 𝐾))) |
25 | 24 | com23 86 | . . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝑀 < 𝐾 → (𝑀 ∈ ℤ → ¬ 𝑀 = 𝐾))) |
26 | 25 | impcom 411 | . . . . . . . . 9 ⊢ ((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ ℤ → ¬ 𝑀 = 𝐾)) |
27 | 26 | imp 410 | . . . . . . . 8 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → ¬ 𝑀 = 𝐾) |
28 | 15, 27 | jctird 530 | . . . . . . 7 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → (1 ≤ 𝑀 → (¬ 𝑀 = 0 ∧ ¬ 𝑀 = 𝐾))) |
29 | 4, 5, 6, 28 | syl21anc 836 | . . . . . 6 ⊢ (𝑀 ∈ (1..^𝐾) → (1 ≤ 𝑀 → (¬ 𝑀 = 0 ∧ ¬ 𝑀 = 𝐾))) |
30 | 3, 29 | mpd 15 | . . . . 5 ⊢ (𝑀 ∈ (1..^𝐾) → (¬ 𝑀 = 0 ∧ ¬ 𝑀 = 𝐾)) |
31 | ioran 981 | . . . . 5 ⊢ (¬ (𝑀 = 0 ∨ 𝑀 = 𝐾) ↔ (¬ 𝑀 = 0 ∧ ¬ 𝑀 = 𝐾)) | |
32 | 30, 31 | sylibr 237 | . . . 4 ⊢ (𝑀 ∈ (1..^𝐾) → ¬ (𝑀 = 0 ∨ 𝑀 = 𝐾)) |
33 | 32 | a1i 11 | . . 3 ⊢ (𝑀 ∈ (0...𝐾) → (𝑀 ∈ (1..^𝐾) → ¬ (𝑀 = 0 ∨ 𝑀 = 𝐾))) |
34 | 33 | con2d 136 | . 2 ⊢ (𝑀 ∈ (0...𝐾) → ((𝑀 = 0 ∨ 𝑀 = 𝐾) → ¬ 𝑀 ∈ (1..^𝐾))) |
35 | 2, 34 | impbid 215 | 1 ⊢ (𝑀 ∈ (0...𝐾) → (¬ 𝑀 ∈ (1..^𝐾) ↔ (𝑀 = 0 ∨ 𝑀 = 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 < clt 10664 ≤ cle 10665 ℤcz 11969 ...cfz 12885 ..^cfzo 13028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 |
This theorem is referenced by: circlemethhgt 32024 |
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