![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nn0disj | GIF version |
Description: The first 𝑁 + 1 elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.) |
Ref | Expression |
---|---|
nn0disj | ⊢ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3120 | . . . . . . 7 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) ↔ (𝑘 ∈ (0...𝑁) ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1)))) | |
2 | 1 | simprbi 260 | . . . . . 6 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) |
3 | eluzle 8261 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝑁 + 1) ≤ 𝑘) | |
4 | 2, 3 | syl 14 | . . . . 5 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → (𝑁 + 1) ≤ 𝑘) |
5 | eluzel2 8254 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝑁 + 1) ∈ ℤ) | |
6 | 2, 5 | syl 14 | . . . . . 6 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → (𝑁 + 1) ∈ ℤ) |
7 | eluzelz 8258 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → 𝑘 ∈ ℤ) | |
8 | 2, 7 | syl 14 | . . . . . 6 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ ℤ) |
9 | zlem1lt 8076 | . . . . . 6 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑁 + 1) ≤ 𝑘 ↔ ((𝑁 + 1) − 1) < 𝑘)) | |
10 | 6, 8, 9 | syl2anc 391 | . . . . 5 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → ((𝑁 + 1) ≤ 𝑘 ↔ ((𝑁 + 1) − 1) < 𝑘)) |
11 | 4, 10 | mpbid 135 | . . . 4 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → ((𝑁 + 1) − 1) < 𝑘) |
12 | 1 | simplbi 259 | . . . . . 6 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ (0...𝑁)) |
13 | elfzle2 8662 | . . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ≤ 𝑁) | |
14 | 12, 13 | syl 14 | . . . . 5 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → 𝑘 ≤ 𝑁) |
15 | 8 | zred 8136 | . . . . . . 7 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ ℝ) |
16 | elfzel2 8658 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑁 ∈ ℤ) | |
17 | 16 | adantr 261 | . . . . . . . . 9 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑁 ∈ ℤ) |
18 | 1, 17 | sylbi 114 | . . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → 𝑁 ∈ ℤ) |
19 | 18 | zred 8136 | . . . . . . 7 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → 𝑁 ∈ ℝ) |
20 | 15, 19 | lenltd 6931 | . . . . . 6 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → (𝑘 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑘)) |
21 | 18 | zcnd 8137 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → 𝑁 ∈ ℂ) |
22 | pncan1 7171 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁) | |
23 | 21, 22 | syl 14 | . . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → ((𝑁 + 1) − 1) = 𝑁) |
24 | 23 | eqcomd 2042 | . . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → 𝑁 = ((𝑁 + 1) − 1)) |
25 | 24 | breq1d 3765 | . . . . . . 7 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → (𝑁 < 𝑘 ↔ ((𝑁 + 1) − 1) < 𝑘)) |
26 | 25 | notbid 591 | . . . . . 6 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → (¬ 𝑁 < 𝑘 ↔ ¬ ((𝑁 + 1) − 1) < 𝑘)) |
27 | 20, 26 | bitrd 177 | . . . . 5 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → (𝑘 ≤ 𝑁 ↔ ¬ ((𝑁 + 1) − 1) < 𝑘)) |
28 | 14, 27 | mpbid 135 | . . . 4 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → ¬ ((𝑁 + 1) − 1) < 𝑘) |
29 | 11, 28 | pm2.21dd 550 | . . 3 ⊢ (𝑘 ∈ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ ∅) |
30 | 29 | ssriv 2943 | . 2 ⊢ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) ⊆ ∅ |
31 | ss0 3251 | . 2 ⊢ (((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) ⊆ ∅ → ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) = ∅) | |
32 | 30, 31 | ax-mp 7 | 1 ⊢ ((0...𝑁) ∩ (ℤ≥‘(𝑁 + 1))) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∩ cin 2910 ⊆ wss 2911 ∅c0 3218 class class class wbr 3755 ‘cfv 4845 (class class class)co 5455 ℂcc 6709 0cc0 6711 1c1 6712 + caddc 6714 < clt 6857 ≤ cle 6858 − cmin 6979 ℤcz 8021 ℤ≥cuz 8249 ...cfz 8644 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-addass 6785 ax-distr 6787 ax-i2m1 6788 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 ax-pre-ltirr 6795 ax-pre-ltwlin 6796 ax-pre-lttrn 6797 ax-pre-ltadd 6799 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 df-0r 6659 df-1r 6660 df-0 6718 df-1 6719 df-r 6721 df-lt 6724 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-sub 6981 df-neg 6982 df-inn 7696 df-n0 7958 df-z 8022 df-uz 8250 df-fz 8645 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |