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Mirrors > Home > MPE Home > Th. List > 4nn0 | Structured version Visualization version GIF version |
Description: 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
4nn0 | ⊢ 4 ∈ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn 11709 | . 2 ⊢ 4 ∈ ℕ | |
2 | 1 | nnnn0i 11894 | 1 ⊢ 4 ∈ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 4c4 11683 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-4 11691 df-n0 11887 |
This theorem is referenced by: 6p5e11 12160 7p5e12 12164 8p5e13 12170 8p7e15 12172 9p5e14 12177 9p6e15 12178 4t3e12 12185 4t4e16 12186 5t5e25 12190 6t4e24 12193 6t5e30 12194 7t3e21 12197 7t5e35 12199 7t7e49 12201 8t3e24 12203 8t4e32 12204 8t5e40 12205 8t6e48 12206 8t7e56 12207 8t8e64 12208 9t5e45 12212 9t6e54 12213 9t7e63 12214 decbin3 12229 fzo0to42pr 13114 4bc3eq4 13678 bpoly4 15403 fsumcube 15404 resin4p 15481 recos4p 15482 ef01bndlem 15527 sin01bnd 15528 cos01bnd 15529 prm23lt5 16141 decexp2 16401 2exp8 16413 2exp16 16414 2expltfac 16416 13prm 16439 19prm 16441 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 resshom 16681 slotsbhcdif 16683 prdsvalstr 16716 oppchomfval 16974 oppcbas 16978 rescbas 17089 rescco 17092 rescabs 17093 catstr 17217 lt6abl 18946 cnfldfun 20487 binom4 25355 dquart 25358 quart1cl 25359 quart1lem 25360 quart1 25361 log2ublem3 25454 log2ub 25455 ppiublem2 25707 bclbnd 25784 bpos1 25787 bposlem8 25795 bposlem9 25796 bpos 25797 2lgslem3a 25900 2lgslem3b 25901 2lgslem3c 25902 2lgslem3d 25903 usgrexmplef 26969 upgr4cycl4dv4e 27892 ex-exp 28157 ex-fac 28158 ex-bc 28159 ex-ind-dvds 28168 hgt750lemd 31819 hgt750lem 31822 hgt750lem2 31823 hgt750leme 31829 tgoldbachgtde 31831 kur14lem9 32359 235t711 39057 ex-decpmul 39058 3cubeslem3l 39163 3cubeslem3r 39164 rmxdioph 39493 inductionexd 40385 amgm4d 40434 wallispi2lem1 42237 wallispi2lem2 42238 wallispi2 42239 stirlinglem3 42242 stirlinglem8 42247 stirlinglem15 42254 smfmullem2 42948 fmtno4 43561 fmtno5lem4 43565 fmtno5 43566 257prm 43570 fmtno4prmfac 43581 fmtno4prmfac193 43582 fmtno4nprmfac193 43583 fmtno4prm 43584 fmtnofz04prm 43586 fmtnole4prm 43587 fmtno5faclem1 43588 fmtno5faclem2 43589 fmtno5faclem3 43590 fmtno5fac 43591 fmtno5nprm 43592 139prmALT 43606 2exp7 43609 127prm 43610 2exp11 43612 m11nprm 43613 3exp4mod41 43628 41prothprmlem2 43630 2exp340mod341 43745 341fppr2 43746 8exp8mod9 43748 nfermltl2rev 43755 |
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