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Mirrors > Home > MPE Home > Th. List > 9nn | Structured version Visualization version GIF version |
Description: 9 is a positive integer. (Contributed by NM, 21-Oct-2012.) |
Ref | Expression |
---|---|
9nn | ⊢ 9 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-9 12181 | . 2 ⊢ 9 = (8 + 1) | |
2 | 8nn 12206 | . . 3 ⊢ 8 ∈ ℕ | |
3 | peano2nn 12123 | . . 3 ⊢ (8 ∈ ℕ → (8 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (8 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2834 | 1 ⊢ 9 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 (class class class)co 7351 1c1 11010 + caddc 11012 ℕcn 12111 8c8 12172 9c9 12173 |
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 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 ax-1cn 11067 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 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 7354 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 |
This theorem is referenced by: 9nn0 12395 9p1e10 12578 10nn 12592 3dvdsdec 16174 19prm 16950 prmlem2 16952 37prm 16953 43prm 16954 83prm 16955 139prm 16956 163prm 16957 317prm 16958 631prm 16959 1259lem1 16963 1259lem2 16964 1259lem3 16965 1259lem4 16966 1259lem5 16967 2503lem3 16971 tsetndx 17193 tsetid 17194 tsetndxnn 17195 topgrpstr 17202 otpsstr 17217 odrngstr 17244 imasvalstr 17293 ipostr 18378 oppgtsetOLD 19092 mgptsetOLD 19866 sratsetOLD 20605 cnfldstr 20751 psrvalstr 21271 eltpsgOLD 22245 indistpsALTOLD 22316 2logb9irr 26097 sqrt2cxp2logb9e3 26101 mcubic 26149 log2cnv 26246 log2tlbnd 26247 log2ublem2 26249 log2ub 26251 bposlem7 26590 ex-cnv 29210 ex-dm 29212 ex-gcd 29230 ex-lcm 29231 ex-prmo 29232 idlsrgstr 32066 hgt750lem2 33077 lcmineqlem23 40446 3lexlogpow2ineq1 40453 3lexlogpow2ineq2 40454 rmydioph 41247 deccarry 45444 257prm 45654 fmtno4nprmfac193 45667 139prmALT 45689 127prm 45692 8exp8mod9 45829 9fppr8 45830 nfermltl8rev 45835 wtgoldbnnsum4prm 45895 bgoldbnnsum3prm 45897 bgoldbtbndlem1 45898 tgblthelfgott 45908 |
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