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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 13145 | . . 3 ⊢ ((𝑀 ∈ (0...𝐾) ∧ ¬ 𝑀 ∈ (1..^𝐾)) → (𝑀 = 0 ∨ 𝑀 = 𝐾)) | |
2 | 1 | ex 415 | . 2 ⊢ (𝑀 ∈ (0...𝐾) → (¬ 𝑀 ∈ (1..^𝐾) → (𝑀 = 0 ∨ 𝑀 = 𝐾))) |
3 | elfzole1 13049 | . . . . . 6 ⊢ (𝑀 ∈ (1..^𝐾) → 1 ≤ 𝑀) | |
4 | elfzolt2 13050 | . . . . . . 7 ⊢ (𝑀 ∈ (1..^𝐾) → 𝑀 < 𝐾) | |
5 | elfzoel2 13040 | . . . . . . 7 ⊢ (𝑀 ∈ (1..^𝐾) → 𝐾 ∈ ℤ) | |
6 | elfzoelz 13041 | . . . . . . 7 ⊢ (𝑀 ∈ (1..^𝐾) → 𝑀 ∈ ℤ) | |
7 | 0lt1 11164 | . . . . . . . . . . 11 ⊢ 0 < 1 | |
8 | breq1 5071 | . . . . . . . . . . 11 ⊢ (𝑀 = 0 → (𝑀 < 1 ↔ 0 < 1)) | |
9 | 7, 8 | mpbiri 260 | . . . . . . . . . 10 ⊢ (𝑀 = 0 → 𝑀 < 1) |
10 | zre 11988 | . . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
11 | 10 | adantl 484 | . . . . . . . . . . 11 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℝ) |
12 | 1red 10644 | . . . . . . . . . . 11 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → 1 ∈ ℝ) | |
13 | 11, 12 | ltnled 10789 | . . . . . . . . . 10 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → (𝑀 < 1 ↔ ¬ 1 ≤ 𝑀)) |
14 | 9, 13 | syl5ib 246 | . . . . . . . . 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 11988 | . . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
17 | ltlen 10743 | . . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (𝑀 < 𝐾 ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≠ 𝑀))) | |
18 | 10, 16, 17 | syl2anr 598 | . . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 < 𝐾 ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≠ 𝑀))) |
19 | necom 3071 | . . . . . . . . . . . . . . 15 ⊢ (𝐾 ≠ 𝑀 ↔ 𝑀 ≠ 𝐾) | |
20 | df-ne 3019 | . . . . . . . . . . . . . . 15 ⊢ (𝑀 ≠ 𝐾 ↔ ¬ 𝑀 = 𝐾) | |
21 | 19, 20 | sylbb 221 | . . . . . . . . . . . . . 14 ⊢ (𝐾 ≠ 𝑀 → ¬ 𝑀 = 𝐾) |
22 | 21 | adantl 484 | . . . . . . . . . . . . 13 ⊢ ((𝑀 ≤ 𝐾 ∧ 𝐾 ≠ 𝑀) → ¬ 𝑀 = 𝐾) |
23 | 18, 22 | syl6bi 255 | . . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 < 𝐾 → ¬ 𝑀 = 𝐾)) |
24 | 23 | ex 415 | . . . . . . . . . . 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 410 | . . . . . . . . 9 ⊢ ((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ ℤ → ¬ 𝑀 = 𝐾)) |
27 | 26 | imp 409 | . . . . . . . 8 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → ¬ 𝑀 = 𝐾) |
28 | 15, 27 | jctird 529 | . . . . . . 7 ⊢ (((𝑀 < 𝐾 ∧ 𝐾 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → (1 ≤ 𝑀 → (¬ 𝑀 = 0 ∧ ¬ 𝑀 = 𝐾))) |
29 | 4, 5, 6, 28 | syl21anc 835 | . . . . . 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 980 | . . . . 5 ⊢ (¬ (𝑀 = 0 ∨ 𝑀 = 𝐾) ↔ (¬ 𝑀 = 0 ∧ ¬ 𝑀 = 𝐾)) | |
32 | 30, 31 | sylibr 236 | . . . 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 214 | 1 ⊢ (𝑀 ∈ (0...𝐾) → (¬ 𝑀 ∈ (1..^𝐾) ↔ (𝑀 = 0 ∨ 𝑀 = 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 < clt 10677 ≤ cle 10678 ℤcz 11984 ...cfz 12895 ..^cfzo 13036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 |
This theorem is referenced by: circlemethhgt 31916 |
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