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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 12946 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁)) | |
2 | 1re 11252 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
3 | zre 12605 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
4 | ltle 11340 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (1 < 𝑁 → 1 ≤ 𝑁)) | |
5 | 2, 3, 4 | sylancr 585 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (1 < 𝑁 → 1 ≤ 𝑁)) |
6 | 5 | imdistani 567 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) |
7 | elnnz1 12631 | . . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) | |
8 | 6, 7 | sylibr 233 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → 𝑁 ∈ ℕ) |
9 | simpr 483 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → 1 < 𝑁) | |
10 | 8, 9 | jca 510 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) → (𝑁 ∈ ℕ ∧ 1 < 𝑁)) |
11 | nnz 12622 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
12 | 11 | anim1i 613 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 1 < 𝑁) → (𝑁 ∈ ℤ ∧ 1 < 𝑁)) |
13 | 10, 12 | impbii 208 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) |
14 | 1, 13 | bitri 274 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2099 class class class wbr 5143 ‘cfv 6543 ℝcr 11145 1c1 11147 < clt 11286 ≤ cle 11287 ℕcn 12255 2c2 12310 ℤcz 12601 ℤ≥cuz 12865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-2 12318 df-n0 12516 df-z 12602 df-uz 12866 |
This theorem is referenced by: eluz2b3 12949 nprmi 16682 ge2nprmge4 16694 pockthlem 16899 prmunb 16908 prmlem1a 17101 sylow3lem6 19623 chtge0 27134 muval1 27155 chtwordi 27178 vma1 27188 mersenne 27250 perfectlem2 27253 lgsne0 27358 chtppilimlem1 27496 padicabvcxp 27655 ostth2lem3 27658 ostth2lem4 27659 ostth2 27660 ostth3 27661 umgr2cwwkdifex 29992 ex-mod 30376 rmspecnonsq 42598 rmspecfund 42600 ltrmxnn0 42641 itgsinexp 45609 wallispilem3 45721 fmtno4prm 47180 perfectALTVlem2 47327 |
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