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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 12247 | . 2 ⊢ 9 ∈ ℕ | |
2 | 1 | nnnn0i 12417 | 1 ⊢ 9 ∈ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 9c9 12211 ℕ0cn0 12409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 ax-1cn 11105 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 |
This theorem is referenced by: deccl 12629 le9lt10 12641 decsucc 12655 9p2e11 12701 9p3e12 12702 9p4e13 12703 9p5e14 12704 9p6e15 12705 9p7e16 12706 9p8e17 12707 9p9e18 12708 9t3e27 12737 9t4e36 12738 9t5e45 12739 9t6e54 12740 9t7e63 12741 9t8e72 12742 9t9e81 12743 sq10e99m1 14157 3dvds2dec 16207 2exp8 16953 19prm 16982 prmlem2 16984 37prm 16985 83prm 16987 139prm 16988 163prm 16989 317prm 16990 631prm 16991 1259lem1 16995 1259lem2 16996 1259lem3 16997 1259lem4 16998 1259lem5 16999 1259prm 17000 2503lem1 17001 2503lem2 17002 2503lem3 17003 2503prm 17004 4001lem1 17005 4001lem2 17006 4001lem3 17007 4001lem4 17008 dsndxntsetndx 17266 unifndxntsetndx 17273 cnfldfunALTOLD 20795 tuslemOLD 23603 setsmsdsOLD 23815 tnglemOLD 23981 tngdsOLD 23996 log2ublem3 26282 log2ub 26283 bposlem8 26623 9p10ne21 29300 dp2lt10 31623 1mhdrd 31655 hgt750lem2 33134 hgt750leme 33140 kur14lem8 33676 60gcd7e1 40429 3exp7 40477 3lexlogpow5ineq1 40478 3lexlogpow5ineq5 40484 aks4d1p1 40500 sqdeccom12 40741 3cubeslem3r 40948 resqrtvalex 41859 imsqrtvalex 41860 fmtno5lem1 45677 fmtno5lem3 45679 fmtno5lem4 45680 fmtno5 45681 257prm 45685 fmtno4prmfac 45696 fmtno4nprmfac193 45698 fmtno5fac 45706 139prmALT 45720 127prm 45723 m11nprm 45725 2exp340mod341 45857 tgblthelfgott 45939 tgoldbachlt 45940 ackval3012 46710 |
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