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Mirrors > Home > MPE Home > Th. List > 9nn0 | Structured version Visualization version GIF version |
Description: 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9nn0 | ⊢ 9 ∈ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn 11724 | . 2 ⊢ 9 ∈ ℕ | |
2 | 1 | nnnn0i 11894 | 1 ⊢ 9 ∈ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 9c9 11688 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-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 |
This theorem is referenced by: deccl 12102 le9lt10 12114 decsucc 12128 9p2e11 12174 9p3e12 12175 9p4e13 12176 9p5e14 12177 9p6e15 12178 9p7e16 12179 9p8e17 12180 9p9e18 12181 9t3e27 12210 9t4e36 12211 9t5e45 12212 9t6e54 12213 9t7e63 12214 9t8e72 12215 9t9e81 12216 sq10e99m1 13615 3dvds2dec 15672 2exp8 16413 19prm 16441 prmlem2 16443 37prm 16444 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 cnfldfun 20487 tuslem 22805 setsmsds 23015 tnglem 23178 tngds 23186 log2ublem3 25454 log2ub 25455 bposlem8 25795 9p10ne21 28177 dp2lt10 30488 1mhdrd 30520 hgt750lem2 31823 hgt750leme 31829 kur14lem8 32358 sqdeccom12 39055 3cubeslem3r 39164 fmtno5lem1 43562 fmtno5lem3 43564 fmtno5lem4 43565 fmtno5 43566 257prm 43570 fmtno4prmfac 43581 fmtno4nprmfac193 43583 fmtno5fac 43591 139prmALT 43606 127prm 43610 m11nprm 43613 2exp340mod341 43745 tgblthelfgott 43827 tgoldbachlt 43828 |
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