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Mirrors > Home > MPE Home > Th. List > 3nn0 | Structured version Visualization version GIF version |
Description: 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
3nn0 | ⊢ 3 ∈ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn 11705 | . 2 ⊢ 3 ∈ ℕ | |
2 | 1 | nnnn0i 11894 | 1 ⊢ 3 ∈ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 3c3 11682 ℕ0cn0 11886 |
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 df-n0 11887 |
This theorem is referenced by: 7p4e11 12163 7p7e14 12166 8p4e12 12169 8p6e14 12171 9p4e13 12176 9p5e14 12177 4t4e16 12186 5t4e20 12189 6t4e24 12193 6t6e36 12195 7t4e28 12198 7t6e42 12200 8t4e32 12204 8t5e40 12205 9t4e36 12211 9t5e45 12212 9t7e63 12214 9t8e72 12215 fz0to3un2pr 12999 4fvwrd4 13017 fldiv4p1lem1div2 13195 expnass 13560 binom3 13575 fac4 13631 4bc2eq6 13679 hash3tr 13838 bpoly3 15402 bpoly4 15403 fsumcube 15404 ef4p 15456 efi4p 15480 resin4p 15481 recos4p 15482 ef01bndlem 15527 sin01bnd 15528 sin01gt0 15533 2exp6 16412 2exp8 16413 2exp16 16414 3exp3 16415 7prm 16434 11prm 16438 13prm 16439 17prm 16440 23prm 16442 prmlem2 16443 37prm 16444 43prm 16445 83prm 16446 139prm 16447 163prm 16448 317prm 16449 631prm 16450 1259lem1 16454 1259lem2 16455 1259lem3 16456 1259lem4 16457 1259lem5 16458 1259prm 16459 2503lem1 16460 2503lem2 16461 2503lem3 16462 2503prm 16463 4001lem1 16464 4001lem2 16465 4001lem3 16466 4001lem4 16467 4001prm 16468 cnfldfun 20487 ressunif 22800 tuslem 22805 tangtx 25020 1cubrlem 25346 dcubic1lem 25348 dcubic2 25349 dcubic1 25350 dcubic 25351 mcubic 25352 cubic2 25353 cubic 25354 binom4 25355 dquartlem2 25357 quart1cl 25359 quart1lem 25360 quart1 25361 quartlem1 25362 quartlem2 25363 quart 25366 log2ublem1 25452 log2ublem3 25454 log2ub 25455 log2le1 25456 birthday 25460 ppiublem2 25707 bclbnd 25784 bpos1 25787 bposlem8 25795 gausslemma2dlem4 25873 2lgslem3b 25901 2lgslem3d 25903 pntlemd 26098 pntlema 26100 pntlemb 26101 pntlemf 26109 pntlemo 26111 pntlem3 26113 tgcgr4 26245 iscgra 26523 isinag 26552 isleag 26561 iseqlg 26581 usgrexmplef 26969 upgr3v3e3cycl 27887 upgr4cycl4dv4e 27892 konigsbergiedgw 27955 konigsberglem1 27959 konigsberglem2 27960 konigsberglem3 27961 konigsberglem4 27962 ex-prmo 28166 threehalves 30519 circlemethhgt 31814 hgt750lemd 31819 hgt750lem 31822 hgt750lem2 31823 hgt750lemb 31827 hgt750lema 31828 hgt750leme 31829 tgoldbachgtde 31831 tgoldbachgtda 31832 tgoldbachgt 31834 cusgracyclt3v 32301 kur14lem8 32358 235t711 39057 ex-decpmul 39058 3cubeslem3l 39163 3cubeslem3r 39164 3cubeslem4 39166 3cubes 39167 jm2.23 39473 jm2.20nn 39474 rmydioph 39491 rmxdioph 39493 expdiophlem2 39499 expdioph 39500 amgm3d 40433 lhe4.4ex1a 40541 fmtno3 43560 fmtno4 43561 fmtno5lem1 43562 fmtno5lem2 43563 fmtno5lem3 43564 fmtno5lem4 43565 fmtno5 43566 257prm 43570 fmtnoprmfac2lem1 43575 fmtno4prmfac 43581 fmtno4prmfac193 43582 fmtno4nprmfac193 43583 fmtno5faclem2 43589 2exp5 43602 139prmALT 43606 31prm 43607 m5prm 43608 127prm 43610 2exp11 43612 m11nprm 43613 mod42tp1mod8 43614 11t31e341 43744 2exp340mod341 43745 341fppr2 43746 8exp8mod9 43748 nfermltl2rev 43755 tgoldbachlt 43828 tgoldbach 43829 zlmodzxzldeplem1 44453 |
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