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| 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 12963 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) | |
| 2 | nngt1ne1 12295 | . . 3 ⊢ (𝑁 ∈ ℕ → (1 < 𝑁 ↔ 𝑁 ≠ 1)) | |
| 3 | 2 | pm5.32i 574 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 1 < 𝑁) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1)) |
| 4 | 1, 3 | bitri 275 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 1c1 11156 < clt 11295 ℕcn 12266 2c2 12321 ℤ≥cuz 12878 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 |
| This theorem is referenced by: 1nuz2 12966 elnn1uz2 12967 nn01to3 12983 relexpuzrel 15091 nno 16419 ncoprmgcdne1b 16687 isprm2 16719 isprm4 16721 rpexp 16759 dfphi2 16811 dvdsprmpweqnn 16923 expnprm 16940 prmirredlem 21483 domnchr 21547 ovolicc1 25551 logbgcd1irr 26837 musum 27234 lgsne0 27379 2sqlem8a 27469 2sqlem8 27470 2sqlem9 27471 2sqcoprm 27479 frgrregord013 30414 ballotlemic 34509 ballotlem1c 34510 signstfveq0a 34591 subfacp1lem3 35187 stoweidlem14 46029 lighneallem3 47594 lighneallem4 47597 eluz2cnn0n1 48428 |
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