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| Mirrors > Home > MPE Home > Th. List > 3nn | Structured version Visualization version GIF version | ||
| Description: 3 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
| Ref | Expression |
|---|---|
| 3nn | ⊢ 3 ∈ ℕ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-3 12330 | . 2 ⊢ 3 = (2 + 1) | |
| 2 | 2nn 12339 | . . 3 ⊢ 2 ∈ ℕ | |
| 3 | peano2nn 12278 | . . 3 ⊢ (2 ∈ ℕ → (2 + 1) ∈ ℕ) | |
| 4 | 2, 3 | ax-mp 5 | . 2 ⊢ (2 + 1) ∈ ℕ |
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ 3 ∈ ℕ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 1c1 11156 + caddc 11158 ℕcn 12266 2c2 12321 3c3 12322 |
| 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-pr 5432 ax-un 7755 ax-1cn 11213 |
| 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-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-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-2 12329 df-3 12330 |
| This theorem is referenced by: 4nn 12349 3nn0 12544 3z 12650 ige3m2fz 13588 fvf1tp 13829 tpf1ofv0 14535 tpf1ofv1 14536 tpf1ofv2 14537 tpfo 14539 f1oun2prg 14956 01sqrexlem7 15287 bpoly4 16095 fsumcube 16096 sin01bnd 16221 egt2lt3 16242 rpnnen2lem2 16251 rpnnen2lem3 16252 rpnnen2lem4 16253 rpnnen2lem9 16258 rpnnen2lem11 16260 5ndvds3 16450 3lcm2e6woprm 16652 3lcm2e6 16769 prmo3 17079 5prm 17146 6nprm 17147 7prm 17148 9nprm 17150 11prm 17152 13prm 17153 17prm 17154 19prm 17155 23prm 17156 prmlem2 17157 37prm 17158 43prm 17159 83prm 17160 139prm 17161 163prm 17162 317prm 17163 631prm 17164 1259lem5 17172 2503lem1 17174 2503lem2 17175 2503lem3 17176 4001lem4 17181 4001prm 17182 mulrndx 17337 mulridx 17338 rngstr 17342 unifndx 17439 unifid 17440 unifndxnn 17441 slotsdifunifndx 17445 lt6abl 19913 sramulrOLD 21182 cnfldstr 21366 cnfldstrOLD 21381 cnfldfunALTOLDOLD 21393 zlmmulrOLD 21534 znmulOLD 21560 opsrmulrOLD 22074 tuslemOLD 24276 tngmulrOLD 24661 tangtx 26547 1cubrlem 26884 1cubr 26885 dcubic1lem 26886 dcubic2 26887 dcubic 26889 mcubic 26890 cubic2 26891 cubic 26892 quartlem3 26902 quart 26904 log2cnv 26987 log2tlbnd 26988 log2ublem1 26989 log2ublem2 26990 log2ub 26992 ppiublem1 27246 ppiub 27248 chtub 27256 bposlem3 27330 bposlem4 27331 bposlem5 27332 bposlem6 27333 bposlem9 27336 lgsdir2lem5 27373 dchrvmasumlem2 27542 dchrvmasumlema 27544 pntleml 27655 tgcgr4 28539 axlowdimlem16 28972 axlowdimlem17 28973 usgrexmpldifpr 29275 upgr3v3e3cycl 30199 ex-cnv 30456 ex-rn 30459 ex-mod 30468 resvmulrOLD 33366 2sqr3minply 33791 fib4 34406 circlevma 34657 circlemethhgt 34658 hgt750lema 34672 sinccvglem 35677 cnndvlem1 36538 mblfinlem3 37666 itg2addnclem2 37679 itg2addnc 37681 hlhilsmulOLD 41947 lcm3un 42016 aks4d1p1 42077 3cubeslem2 42696 3cubeslem3r 42698 3cubes 42701 rmydioph 43026 rmxdioph 43028 expdiophlem2 43034 expdioph 43035 amgm3d 44212 lhe4.4ex1a 44348 257prm 47548 fmtno4prmfac193 47560 fmtno4nprmfac193 47561 3ndvds4 47582 139prmALT 47583 31prm 47584 127prm 47586 41prothprm 47606 341fppr2 47721 nfermltl2rev 47730 wtgoldbnnsum4prm 47789 bgoldbnnsum3prm 47791 bgoldbtbndlem1 47792 tgoldbach 47804 grtriclwlk3 47912 gpg3kgrtriexlem2 48040 gpg3kgrtriexlem5 48043 gpg3kgrtriexlem6 48044 gpg3kgrtriex 48045 gpg5grlic 48047 |
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