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Mirrors > Home > ILE Home > Th. List > fzn | 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 | fznlem 9814 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 → (𝑀...𝑁) = ∅)) | |
2 | neq0r 3372 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) → ¬ (𝑀...𝑁) = ∅) | |
3 | simpr 109 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀...𝑁) = ∅) → (𝑀...𝑁) = ∅) | |
4 | 2, 3 | nsyl3 615 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀...𝑁) = ∅) → ¬ ∃𝑥 𝑥 ∈ (𝑀...𝑁)) |
5 | fzm 9811 | . . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ≥‘𝑀)) | |
6 | eluz 9332 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) | |
7 | 5, 6 | syl5rbb 192 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ ∃𝑥 𝑥 ∈ (𝑀...𝑁))) |
8 | 7 | adantr 274 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀...𝑁) = ∅) → (𝑀 ≤ 𝑁 ↔ ∃𝑥 𝑥 ∈ (𝑀...𝑁))) |
9 | 4, 8 | mtbird 662 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀...𝑁) = ∅) → ¬ 𝑀 ≤ 𝑁) |
10 | zltnle 9093 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁)) | |
11 | 10 | ancoms 266 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁)) |
12 | 11 | adantr 274 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀...𝑁) = ∅) → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁)) |
13 | 9, 12 | mpbird 166 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀...𝑁) = ∅) → 𝑁 < 𝑀) |
14 | 13 | ex 114 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀...𝑁) = ∅ → 𝑁 < 𝑀)) |
15 | 1, 14 | impbid 128 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∅c0 3358 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 < clt 7793 ≤ cle 7794 ℤcz 9047 ℤ≥cuz 9319 ...cfz 9783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 |
This theorem is referenced by: fz1n 9817 fz10 9819 fzsuc2 9852 fzm1 9873 fzon 9936 exfzdc 10010 fzfig 10196 uzsinds 10208 hashfzp1 10563 fisumrev2 11208 isumsplit 11253 arisum2 11261 cvgratnnlemseq 11288 |
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