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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 13517 | . . . 4 ⊢ ((𝑀...𝑁) ≠ ∅ ↔ 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluz 12838 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) | |
3 | 1, 2 | bitrid 282 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀...𝑁) ≠ ∅ ↔ 𝑀 ≤ 𝑁)) |
4 | zre 12564 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 12564 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | lenlt 11294 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀)) | |
7 | 4, 5, 6 | syl2an 596 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀)) |
8 | 3, 7 | bitr2d 279 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑁 < 𝑀 ↔ (𝑀...𝑁) ≠ ∅)) |
9 | 8 | necon4bbid 2982 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∅c0 4322 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 ℝcr 11111 < clt 11250 ≤ cle 11251 ℤcz 12560 ℤ≥cuz 12824 ...cfz 13486 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-neg 11449 df-z 12561 df-uz 12825 df-fz 13487 |
This theorem is referenced by: fz1n 13521 fz10 13524 fzsuc2 13561 fzm1 13583 fzon 13655 hashfzp1 14393 isumsplit 15788 arisum2 15809 risefall0lem 15972 prmreclem4 16854 prmreclem5 16855 ppi1 26675 cht1 26676 ppiublem2 26713 lgsdir2lem3 26837 wlkv0 28946 chtvalz 33710 fz0n 34775 poimirlem10 36590 poimirlem23 36603 poimirlem28 36608 fdc 36705 mettrifi 36717 sticksstones11 41064 metakunt24 41100 fzisoeu 44095 fzdifsuc2 44105 |
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