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Mirrors > Home > ILE Home > Th. List > uzdisj | GIF version |
Description: The first 𝑁 elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
uzdisj | ⊢ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3303 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) ↔ (𝑘 ∈ (𝑀...(𝑁 − 1)) ∧ 𝑘 ∈ (ℤ≥‘𝑁))) | |
2 | 1 | simprbi 273 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → 𝑘 ∈ (ℤ≥‘𝑁)) |
3 | eluzle 9472 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → 𝑁 ≤ 𝑘) | |
4 | 2, 3 | syl 14 | . . . . 5 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝑘) |
5 | eluzel2 9465 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → 𝑁 ∈ ℤ) | |
6 | 2, 5 | syl 14 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → 𝑁 ∈ ℤ) |
7 | eluzelz 9469 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → 𝑘 ∈ ℤ) | |
8 | 2, 7 | syl 14 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → 𝑘 ∈ ℤ) |
9 | zlem1lt 9241 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑁 ≤ 𝑘 ↔ (𝑁 − 1) < 𝑘)) | |
10 | 6, 8, 9 | syl2anc 409 | . . . . 5 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → (𝑁 ≤ 𝑘 ↔ (𝑁 − 1) < 𝑘)) |
11 | 4, 10 | mpbid 146 | . . . 4 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → (𝑁 − 1) < 𝑘) |
12 | 1 | simplbi 272 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → 𝑘 ∈ (𝑀...(𝑁 − 1))) |
13 | elfzle2 9957 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...(𝑁 − 1)) → 𝑘 ≤ (𝑁 − 1)) | |
14 | 12, 13 | syl 14 | . . . . 5 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → 𝑘 ≤ (𝑁 − 1)) |
15 | 8 | zred 9307 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → 𝑘 ∈ ℝ) |
16 | peano2zm 9223 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
17 | 6, 16 | syl 14 | . . . . . . 7 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → (𝑁 − 1) ∈ ℤ) |
18 | 17 | zred 9307 | . . . . . 6 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → (𝑁 − 1) ∈ ℝ) |
19 | 15, 18 | lenltd 8010 | . . . . 5 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → (𝑘 ≤ (𝑁 − 1) ↔ ¬ (𝑁 − 1) < 𝑘)) |
20 | 14, 19 | mpbid 146 | . . . 4 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → ¬ (𝑁 − 1) < 𝑘) |
21 | 11, 20 | pm2.21dd 610 | . . 3 ⊢ (𝑘 ∈ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) → 𝑘 ∈ ∅) |
22 | 21 | ssriv 3144 | . 2 ⊢ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) ⊆ ∅ |
23 | ss0 3447 | . 2 ⊢ (((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) ⊆ ∅ → ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) = ∅) | |
24 | 22, 23 | ax-mp 5 | 1 ⊢ ((𝑀...(𝑁 − 1)) ∩ (ℤ≥‘𝑁)) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∩ cin 3113 ⊆ wss 3114 ∅c0 3407 class class class wbr 3979 ‘cfv 5185 (class class class)co 5839 1c1 7748 < clt 7927 ≤ cle 7928 − cmin 8063 ℤcz 9185 ℤ≥cuz 9460 ...cfz 9938 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-addcom 7847 ax-addass 7849 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-0id 7855 ax-rnegex 7856 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-ltwlin 7860 ax-pre-lttrn 7861 ax-pre-ltadd 7863 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-br 3980 df-opab 4041 df-mpt 4042 df-id 4268 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 df-inn 8852 df-n0 9109 df-z 9186 df-uz 9461 df-fz 9939 |
This theorem is referenced by: 2prm 12053 |
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