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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 12364 | . 2 ⊢ 9 ∈ ℕ | |
| 2 | 1 | nnnn0i 12534 | 1 ⊢ 9 ∈ ℕ0 |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 9c9 12328 ℕ0cn0 12526 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-1cn 11213 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 |
| This theorem is referenced by: deccl 12748 le9lt10 12760 decsucc 12774 9p2e11 12820 9p3e12 12821 9p4e13 12822 9p5e14 12823 9p6e15 12824 9p7e16 12825 9p8e17 12826 9p9e18 12827 9t3e27 12856 9t4e36 12857 9t5e45 12858 9t6e54 12859 9t7e63 12860 9t8e72 12861 9t9e81 12862 sq10e99m1 14304 3dvds2dec 16370 2exp8 17126 19prm 17155 prmlem2 17157 37prm 17158 83prm 17160 139prm 17161 163prm 17162 317prm 17163 631prm 17164 1259lem1 17168 1259lem2 17169 1259lem3 17170 1259lem4 17171 1259lem5 17172 1259prm 17173 2503lem1 17174 2503lem2 17175 2503lem3 17176 2503prm 17177 4001lem1 17178 4001lem2 17179 4001lem3 17180 4001lem4 17181 dsndxntsetndx 17437 unifndxntsetndx 17444 cnfldfunALTOLDOLD 21393 tuslemOLD 24276 setsmsdsOLD 24488 tnglemOLD 24654 tngdsOLD 24669 log2ublem3 26991 log2ub 26992 bposlem8 27335 9p10ne21 30489 dp2lt10 32866 1mhdrd 32898 hgt750lem2 34667 hgt750leme 34673 kur14lem8 35218 60gcd7e1 42006 3exp7 42054 3lexlogpow5ineq1 42055 3lexlogpow5ineq5 42061 aks4d1p1 42077 sqdeccom12 42324 sum9cubes 42682 3cubeslem3r 42698 resqrtvalex 43658 imsqrtvalex 43659 fmtno5lem1 47540 fmtno5lem3 47542 fmtno5lem4 47543 fmtno5 47544 257prm 47548 fmtno4prmfac 47559 fmtno4nprmfac193 47561 fmtno5fac 47569 139prmALT 47583 127prm 47586 m11nprm 47588 2exp340mod341 47720 tgblthelfgott 47802 tgoldbachlt 47803 ackval3012 48613 |
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