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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 11690 | . 2 ⊢ 3 = (2 + 1) | |
2 | 2nn 11699 | . . 3 ⊢ 2 ∈ ℕ | |
3 | peano2nn 11639 | . . 3 ⊢ (2 ∈ ℕ → (2 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (2 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2909 | 1 ⊢ 3 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7145 1c1 10527 + caddc 10529 ℕcn 11627 2c2 11681 3c3 11682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-1cn 10584 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-nn 11628 df-2 11689 df-3 11690 |
This theorem is referenced by: 4nn 11709 3nn0 11904 3z 12004 ige3m2fz 12921 f1oun2prg 14269 sqrlem7 14598 bpoly4 15403 fsumcube 15404 sin01bnd 15528 egt2lt3 15549 rpnnen2lem2 15558 rpnnen2lem3 15559 rpnnen2lem4 15560 rpnnen2lem9 15565 rpnnen2lem11 15567 3lcm2e6woprm 15949 3lcm2e6 16062 prmo3 16367 5prm 16432 6nprm 16433 7prm 16434 9nprm 16436 11prm 16438 13prm 16439 17prm 16440 19prm 16441 23prm 16442 prmlem2 16443 37prm 16444 43prm 16445 83prm 16446 139prm 16447 163prm 16448 317prm 16449 631prm 16450 1259lem5 16458 2503lem1 16460 2503lem2 16461 2503lem3 16462 4001lem4 16467 4001prm 16468 mulrndx 16605 mulrid 16606 rngstr 16609 ressmulr 16615 unifndx 16667 unifid 16668 lt6abl 18946 sramulr 19883 opsrmulr 20191 cnfldstr 20477 cnfldfun 20487 zlmmulr 20597 znmul 20618 ressunif 22800 tuslem 22805 tngmulr 23182 tangtx 25020 1cubrlem 25346 1cubr 25347 dcubic1lem 25348 dcubic2 25349 dcubic 25351 mcubic 25352 cubic2 25353 cubic 25354 quartlem3 25364 quart 25366 log2cnv 25450 log2tlbnd 25451 log2ublem1 25452 log2ublem2 25453 log2ub 25455 ppiublem1 25706 ppiub 25708 chtub 25716 bposlem3 25790 bposlem4 25791 bposlem5 25792 bposlem6 25793 bposlem9 25796 lgsdir2lem5 25833 dchrvmasumlem2 26002 dchrvmasumlema 26004 pntleml 26115 tgcgr4 26245 axlowdimlem16 26671 axlowdimlem17 26672 usgrexmpldifpr 26968 upgr3v3e3cycl 27887 ex-cnv 28144 ex-rn 28147 ex-mod 28156 resvmulr 30836 fib4 31562 circlevma 31813 circlemethhgt 31814 hgt750lema 31828 sinccvglem 32813 cnndvlem1 33774 mblfinlem3 34813 itg2addnclem2 34826 itg2addnclem3 34827 itg2addnc 34828 hlhilsmul 38959 3cubeslem2 39162 3cubeslem3r 39164 3cubes 39167 rmydioph 39491 rmxdioph 39493 expdiophlem2 39499 expdioph 39500 amgm3d 40433 lhe4.4ex1a 40541 257prm 43570 fmtno4prmfac193 43582 fmtno4nprmfac193 43583 3ndvds4 43605 139prmALT 43606 31prm 43607 127prm 43610 41prothprm 43631 341fppr2 43746 nfermltl2rev 43755 wtgoldbnnsum4prm 43814 bgoldbnnsum3prm 43816 bgoldbtbndlem1 43817 tgoldbach 43829 |
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