Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11704 | . 2 ⊢ 3 ∈ ℕ | |
2 | 1 | nnnn0i 11893 | 1 ⊢ 3 ∈ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 3c3 11681 ℕ0cn0 11885 |
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 df-n0 11886 |
This theorem is referenced by: 7p4e11 12162 7p7e14 12165 8p4e12 12168 8p6e14 12170 9p4e13 12175 9p5e14 12176 4t4e16 12185 5t4e20 12188 6t4e24 12192 6t6e36 12194 7t4e28 12197 7t6e42 12199 8t4e32 12203 8t5e40 12204 9t4e36 12210 9t5e45 12211 9t7e63 12213 9t8e72 12214 fz0to3un2pr 12997 4fvwrd4 13015 fldiv4p1lem1div2 13193 expnass 13558 binom3 13573 fac4 13629 4bc2eq6 13677 hash3tr 13836 bpoly3 15400 bpoly4 15401 fsumcube 15402 ef4p 15454 efi4p 15478 resin4p 15479 recos4p 15480 ef01bndlem 15525 sin01bnd 15526 sin01gt0 15531 2exp6 16410 2exp8 16411 2exp16 16412 3exp3 16413 7prm 16432 11prm 16436 13prm 16437 17prm 16438 23prm 16440 prmlem2 16441 37prm 16442 43prm 16443 83prm 16444 139prm 16445 163prm 16446 317prm 16447 631prm 16448 1259lem1 16452 1259lem2 16453 1259lem3 16454 1259lem4 16455 1259lem5 16456 1259prm 16457 2503lem1 16458 2503lem2 16459 2503lem3 16460 2503prm 16461 4001lem1 16462 4001lem2 16463 4001lem3 16464 4001lem4 16465 4001prm 16466 cnfldfun 20485 ressunif 22798 tuslem 22803 tangtx 25018 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 25451 log2ublem3 25453 log2ub 25454 log2le1 25455 birthday 25459 ppiublem2 25706 bclbnd 25783 bpos1 25786 bposlem8 25794 gausslemma2dlem4 25872 2lgslem3b 25900 2lgslem3d 25902 pntlemd 26097 pntlema 26099 pntlemb 26100 pntlemf 26108 pntlemo 26110 pntlem3 26112 tgcgr4 26244 iscgra 26522 isinag 26551 isleag 26560 iseqlg 26580 usgrexmplef 26968 upgr3v3e3cycl 27886 upgr4cycl4dv4e 27891 konigsbergiedgw 27954 konigsberglem1 27958 konigsberglem2 27959 konigsberglem3 27960 konigsberglem4 27961 ex-prmo 28165 threehalves 30518 circlemethhgt 31813 hgt750lemd 31818 hgt750lem 31821 hgt750lem2 31822 hgt750lemb 31826 hgt750lema 31827 hgt750leme 31828 tgoldbachgtde 31830 tgoldbachgtda 31831 tgoldbachgt 31833 cusgracyclt3v 32300 kur14lem8 32357 235t711 39055 ex-decpmul 39056 3cubeslem3l 39161 3cubeslem3r 39162 3cubeslem4 39164 3cubes 39165 jm2.23 39471 jm2.20nn 39472 rmydioph 39489 rmxdioph 39491 expdiophlem2 39497 expdioph 39498 amgm3d 40430 lhe4.4ex1a 40538 fmtno3 43590 fmtno4 43591 fmtno5lem1 43592 fmtno5lem2 43593 fmtno5lem3 43594 fmtno5lem4 43595 fmtno5 43596 257prm 43600 fmtnoprmfac2lem1 43605 fmtno4prmfac 43611 fmtno4prmfac193 43612 fmtno4nprmfac193 43613 fmtno5faclem2 43619 2exp5 43632 139prmALT 43636 31prm 43637 m5prm 43638 127prm 43640 2exp11 43642 m11nprm 43643 mod42tp1mod8 43644 11t31e341 43774 2exp340mod341 43775 341fppr2 43776 8exp8mod9 43778 nfermltl2rev 43785 tgoldbachlt 43858 tgoldbach 43859 zlmodzxzldeplem1 44483 |
Copyright terms: Public domain | W3C validator |