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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 9570 | . 2 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ≤ 𝐾) | |
2 | eluzel2 9563 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ∈ ℤ) | |
3 | elfzel1 10054 | . . . . 5 ⊢ (𝐾 ∈ (𝑀...(𝑁 − 1)) → 𝑀 ∈ ℤ) | |
4 | elfzm11 10121 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...(𝑁 − 1)) ↔ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁))) | |
5 | simp3 1001 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁) → 𝐾 < 𝑁) | |
6 | 4, 5 | biimtrdi 163 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...(𝑁 − 1)) → 𝐾 < 𝑁)) |
7 | 6 | impancom 260 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (𝑀...(𝑁 − 1))) → (𝑁 ∈ ℤ → 𝐾 < 𝑁)) |
8 | 3, 7 | mpancom 422 | . . . 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 9567 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝐾 ∈ ℤ) | |
11 | zltnle 9329 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐾)) | |
12 | 10, 2, 11 | syl2anc 411 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → (𝐾 < 𝑁 ↔ ¬ 𝑁 ≤ 𝐾)) |
13 | 9, 12 | sylibd 149 | . 2 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → (𝐾 ∈ (𝑀...(𝑁 − 1)) → ¬ 𝑁 ≤ 𝐾)) |
14 | 1, 13 | mt2d 626 | 1 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 ∈ wcel 2160 class class class wbr 4018 ‘cfv 5235 (class class class)co 5896 1c1 7842 < clt 8022 ≤ cle 8023 − cmin 8158 ℤcz 9283 ℤ≥cuz 9558 ...cfz 10038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-ltadd 7957 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-inn 8950 df-n0 9207 df-z 9284 df-uz 9559 df-fz 10039 |
This theorem is referenced by: sumrbdclem 11417 prodrbdclem 11611 |
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