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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 12771 | . . . 4 ⊢ ((𝑀...𝑁) ≠ ∅ ↔ 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluz 12107 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) | |
3 | 1, 2 | syl5bb 284 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀...𝑁) ≠ ∅ ↔ 𝑀 ≤ 𝑁)) |
4 | zre 11835 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 11835 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | lenlt 10568 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀)) | |
7 | 4, 5, 6 | syl2an 595 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀)) |
8 | 3, 7 | bitr2d 281 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑁 < 𝑀 ↔ (𝑀...𝑁) ≠ ∅)) |
9 | 8 | necon4bbid 3024 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2080 ≠ wne 2983 ∅c0 4213 class class class wbr 4964 ‘cfv 6228 (class class class)co 7019 ℝcr 10385 < clt 10524 ≤ cle 10525 ℤcz 11831 ℤ≥cuz 12093 ...cfz 12742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-pre-lttri 10460 ax-pre-lttrn 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-id 5351 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-ov 7022 df-oprab 7023 df-mpo 7024 df-1st 7548 df-2nd 7549 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-neg 10722 df-z 11832 df-uz 12094 df-fz 12743 |
This theorem is referenced by: fz1n 12775 fz10 12778 fzsuc2 12815 fzm1 12837 fzon 12908 hashfzp1 13640 isumsplit 15028 arisum2 15049 risefall0lem 15213 prmreclem4 16084 prmreclem5 16085 ppi1 25423 cht1 25424 ppiublem2 25461 lgsdir2lem3 25585 wlkv0 27115 chtvalz 31509 fz0n 32564 poimirlem10 34446 poimirlem23 34459 poimirlem28 34464 fdc 34565 mettrifi 34577 fzisoeu 41121 fzdifsuc2 41131 |
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