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Mirrors > Home > ILE Home > Th. List > uznfz | GIF version |
Description: Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.) |
Ref | Expression |
---|---|
uznfz | ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzle 9338 | . 2 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ≤ 𝐾) | |
2 | eluzel2 9331 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ∈ ℤ) | |
3 | elfzel1 9805 | . . . . 5 ⊢ (𝐾 ∈ (𝑀...(𝑁 − 1)) → 𝑀 ∈ ℤ) | |
4 | elfzm11 9871 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...(𝑁 − 1)) ↔ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁))) | |
5 | simp3 983 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁) → 𝐾 < 𝑁) | |
6 | 4, 5 | syl6bi 162 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...(𝑁 − 1)) → 𝐾 < 𝑁)) |
7 | 6 | impancom 258 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (𝑀...(𝑁 − 1))) → (𝑁 ∈ ℤ → 𝐾 < 𝑁)) |
8 | 3, 7 | mpancom 418 | . . . 4 ⊢ (𝐾 ∈ (𝑀...(𝑁 − 1)) → (𝑁 ∈ ℤ → 𝐾 < 𝑁)) |
9 | 2, 8 | syl5com 29 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → (𝐾 ∈ (𝑀...(𝑁 − 1)) → 𝐾 < 𝑁)) |
10 | eluzelz 9335 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝐾 ∈ ℤ) | |
11 | zltnle 9100 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐾)) | |
12 | 10, 2, 11 | syl2anc 408 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → (𝐾 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐾)) |
13 | 9, 12 | sylibd 148 | . 2 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → (𝐾 ∈ (𝑀...(𝑁 − 1)) → ¬ 𝑁 ≤ 𝐾)) |
14 | 1, 13 | mt2d 614 | 1 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 ∈ wcel 1480 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 1c1 7621 < clt 7800 ≤ cle 7801 − cmin 7933 ℤcz 9054 ℤ≥cuz 9326 ...cfz 9790 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 |
This theorem is referenced by: sumrbdclem 11146 prodrbdclem 11340 |
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