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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 11230 | . 2 ⊢ 9 ∈ ℕ | |
2 | 1 | nnnn0i 11338 | 1 ⊢ 9 ∈ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 9c9 11115 ℕ0cn0 11330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-1cn 10032 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 |
This theorem is referenced by: deccl 11550 le9lt10 11567 declecOLD 11582 decsucc 11588 decsuccOLD 11589 9p2e11 11657 9p2e11OLD 11658 9p3e12 11659 9p4e13 11660 9p5e14 11661 9p6e15 11662 9p7e16 11663 9p8e17 11664 9p9e18 11665 9t3e27 11702 9t4e36 11703 9t5e45 11704 9t6e54 11705 9t7e63 11706 9t8e72 11707 9t9e81 11708 sq10e99m1 13089 sq10e99m1OLD 13092 3dvds2dec 15103 3dvds2decOLD 15104 2exp8 15843 19prm 15872 prmlem2 15874 37prm 15875 83prm 15877 139prm 15878 163prm 15879 317prm 15880 631prm 15881 1259lem1 15885 1259lem2 15886 1259lem3 15887 1259lem4 15888 1259lem5 15889 1259prm 15890 2503lem1 15891 2503lem2 15892 2503lem3 15893 2503prm 15894 4001lem1 15895 4001lem2 15896 4001lem3 15897 4001lem4 15898 cnfldfun 19806 tuslem 22118 setsmsds 22328 tnglem 22491 tngds 22499 log2ublem3 24720 log2ub 24721 bposlem8 25061 dp2lt10 29719 1mhdrd 29752 hgt750lem2 30858 hgt750leme 30864 kur14lem8 31321 fmtno5lem1 41790 fmtno5lem3 41792 fmtno5lem4 41793 fmtno5 41794 257prm 41798 fmtno4prmfac 41809 fmtno4prmfac193 41810 fmtno4nprmfac193 41811 fmtno5fac 41819 139prmALT 41836 127prm 41840 m11nprm 41843 tgblthelfgott 42028 tgoldbachlt 42029 tgblthelfgottOLD 42034 tgoldbachltOLD 42035 |
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