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Mirrors > Home > MPE Home > Th. List > fzn | Structured version Visualization version GIF version |
Description: A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.) |
Ref | Expression |
---|---|
fzn | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzn0 13410 | . . . 4 ⊢ ((𝑀...𝑁) ≠ ∅ ↔ 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluz 12736 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) | |
3 | 1, 2 | bitrid 283 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀...𝑁) ≠ ∅ ↔ 𝑀 ≤ 𝑁)) |
4 | zre 12462 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 12462 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | lenlt 11192 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀)) | |
7 | 4, 5, 6 | syl2an 597 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀)) |
8 | 3, 7 | bitr2d 280 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑁 < 𝑀 ↔ (𝑀...𝑁) ≠ ∅)) |
9 | 8 | necon4bbid 2984 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∅c0 4281 class class class wbr 5104 ‘cfv 6494 (class class class)co 7352 ℝcr 11009 < clt 11148 ≤ cle 11149 ℤcz 12458 ℤ≥cuz 12722 ...cfz 13379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-pre-lttri 11084 ax-pre-lttrn 11085 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7914 df-2nd 7915 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-neg 11347 df-z 12459 df-uz 12723 df-fz 13380 |
This theorem is referenced by: fz1n 13414 fz10 13417 fzsuc2 13454 fzm1 13476 fzon 13548 hashfzp1 14285 isumsplit 15685 arisum2 15706 risefall0lem 15869 prmreclem4 16751 prmreclem5 16752 ppi1 26465 cht1 26466 ppiublem2 26503 lgsdir2lem3 26627 wlkv0 28428 chtvalz 33046 fz0n 34113 poimirlem10 36020 poimirlem23 36033 poimirlem28 36038 fdc 36136 mettrifi 36148 sticksstones11 40496 metakunt24 40532 fzisoeu 43433 fzdifsuc2 43443 |
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