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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 11689 | . 2 ⊢ 3 = (2 + 1) | |
2 | 2nn 11698 | . . 3 ⊢ 2 ∈ ℕ | |
3 | peano2nn 11638 | . . 3 ⊢ (2 ∈ ℕ → (2 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (2 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2906 | 1 ⊢ 3 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7145 1c1 10526 + caddc 10528 ℕcn 11626 2c2 11680 3c3 11681 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-1cn 10583 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-nn 11627 df-2 11688 df-3 11689 |
This theorem is referenced by: 4nn 11708 3nn0 11903 3z 12003 ige3m2fz 12919 f1oun2prg 14267 sqrlem7 14596 bpoly4 15401 fsumcube 15402 sin01bnd 15526 egt2lt3 15547 rpnnen2lem2 15556 rpnnen2lem3 15557 rpnnen2lem4 15558 rpnnen2lem9 15563 rpnnen2lem11 15565 3lcm2e6woprm 15947 3lcm2e6 16060 prmo3 16365 5prm 16430 6nprm 16431 7prm 16432 9nprm 16434 11prm 16436 13prm 16437 17prm 16438 19prm 16439 23prm 16440 prmlem2 16441 37prm 16442 43prm 16443 83prm 16444 139prm 16445 163prm 16446 317prm 16447 631prm 16448 1259lem5 16456 2503lem1 16458 2503lem2 16459 2503lem3 16460 4001lem4 16465 4001prm 16466 mulrndx 16603 mulrid 16604 rngstr 16607 ressmulr 16613 unifndx 16665 unifid 16666 lt6abl 18944 sramulr 19881 opsrmulr 20189 cnfldstr 20475 cnfldfun 20485 zlmmulr 20595 znmul 20616 ressunif 22798 tuslem 22803 tngmulr 23180 tangtx 25018 1cubrlem 25346 1cubr 25347 dcubic1lem 25348 dcubic2 25349 dcubic 25351 mcubic 25352 cubic2 25353 cubic 25354 quartlem3 25364 quart 25366 log2cnv 25449 log2tlbnd 25450 log2ublem1 25451 log2ublem2 25452 log2ub 25454 ppiublem1 25705 ppiub 25707 chtub 25715 bposlem3 25789 bposlem4 25790 bposlem5 25791 bposlem6 25792 bposlem9 25795 lgsdir2lem5 25832 dchrvmasumlem2 26001 dchrvmasumlema 26003 pntleml 26114 tgcgr4 26244 axlowdimlem16 26670 axlowdimlem17 26671 usgrexmpldifpr 26967 upgr3v3e3cycl 27886 ex-cnv 28143 ex-rn 28146 ex-mod 28155 resvmulr 30835 fib4 31561 circlevma 31812 circlemethhgt 31813 hgt750lema 31827 sinccvglem 32812 cnndvlem1 33773 mblfinlem3 34812 itg2addnclem2 34825 itg2addnclem3 34826 itg2addnc 34827 hlhilsmul 38957 3cubeslem2 39160 3cubeslem3r 39162 3cubes 39165 rmydioph 39489 rmxdioph 39491 expdiophlem2 39497 expdioph 39498 amgm3d 40430 lhe4.4ex1a 40538 257prm 43600 fmtno4prmfac193 43612 fmtno4nprmfac193 43613 3ndvds4 43635 139prmALT 43636 31prm 43637 127prm 43640 41prothprm 43661 341fppr2 43776 nfermltl2rev 43785 wtgoldbnnsum4prm 43844 bgoldbnnsum3prm 43846 bgoldbtbndlem1 43847 tgoldbach 43859 |
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