![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eluz2b3 | 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 |
---|---|
eluz2b3 | ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2b2 11965 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) | |
2 | nngt1ne1 11250 | . . 3 ⊢ (𝑁 ∈ ℕ → (1 < 𝑁 ↔ 𝑁 ≠ 1)) | |
3 | 2 | pm5.32i 558 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 1 < 𝑁) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1)) |
4 | 1, 3 | bitri 264 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 ∈ wcel 2145 ≠ wne 2943 class class class wbr 4787 ‘cfv 6032 1c1 10140 < clt 10277 ℕcn 11223 2c2 11273 ℤ≥cuz 11889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-er 7897 df-en 8111 df-dom 8112 df-sdom 8113 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-nn 11224 df-2 11282 df-n0 11496 df-z 11581 df-uz 11890 |
This theorem is referenced by: 1nuz2 11968 elnn1uz2 11969 nn01to3 11985 relexpuzrel 14001 nno 15307 ncoprmgcdne1b 15572 isprm2 15603 isprm4 15605 rpexp 15640 dfphi2 15687 dvdsprmpweqnn 15797 expnprm 15814 prmirredlem 20057 domnchr 20096 ovolicc1 23505 musum 25139 lgsne0 25282 2sqlem8a 25372 2sqlem8 25373 2sqlem9 25374 frgrregord013 27595 2sqcoprm 29988 ballotlemic 30909 ballotlem1c 30910 signstfveq0a 30994 subfacp1lem3 31503 stoweidlem14 40749 lighneallem3 42053 lighneallem4 42056 eluz2cnn0n1 42830 |
Copyright terms: Public domain | W3C validator |