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Mirrors > Home > MPE Home > Th. List > eluz2b2 | Structured version Visualization version GIF version |
Description: Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.) |
Ref | Expression |
---|---|
eluz2b2 | ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2b1 12066 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁)) | |
2 | 1re 10376 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
3 | zre 11732 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
4 | ltle 10465 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (1 < 𝑁 → 1 ≤ 𝑁)) | |
5 | 2, 3, 4 | sylancr 581 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (1 < 𝑁 → 1 ≤ 𝑁)) |
6 | 5 | imdistani 564 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) |
7 | elnnz1 11755 | . . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) | |
8 | 6, 7 | sylibr 226 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → 𝑁 ∈ ℕ) |
9 | simpr 479 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → 1 < 𝑁) | |
10 | 8, 9 | jca 507 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → (𝑁 ∈ ℕ ∧ 1 < 𝑁)) |
11 | nnz 11751 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
12 | 11 | anim1i 608 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 1 < 𝑁) → (𝑁 ∈ ℤ ∧ 1 < 𝑁)) |
13 | 10, 12 | impbii 201 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) |
14 | 1, 13 | bitri 267 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2106 class class class wbr 4886 ‘cfv 6135 ℝcr 10271 1c1 10273 < clt 10411 ≤ cle 10412 ℕcn 11374 2c2 11430 ℤcz 11728 ℤ≥cuz 11992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-uz 11993 |
This theorem is referenced by: eluz2b3 12069 nprmi 15807 pockthlem 16013 prmunb 16022 prmlem1a 16212 odcau 18403 sylow3lem6 18431 gexexlem 18641 wilthlem1 25246 wilth 25249 chtge0 25290 isppw 25292 muval1 25311 chtwordi 25334 vma1 25344 fsumvma2 25391 chpval2 25395 chpchtsum 25396 chpub 25397 mersenne 25404 perfect1 25405 perfectlem2 25407 lgsne0 25512 2sqblem 25608 chtppilimlem1 25614 rplogsumlem2 25626 rpvmasumlem 25628 dchrisum0flblem2 25650 padicabvcxp 25773 ostth2lem3 25776 ostth2lem4 25777 ostth2 25778 ostth3 25779 umgr2cwwkdifex 27463 ex-mod 27881 rmspecnonsq 38413 rmspecfund 38415 ltrmxnn0 38457 itgsinexp 41080 wallispilem3 41193 fmtno4prm 42490 perfectALTVlem2 42638 |
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