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Mirrors > Home > ILE Home > Th. List > nn0ge0 | Unicode version |
Description: A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nn0ge0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 8290 | . 2 | |
2 | nnre 8046 | . . . 4 | |
3 | nngt0 8064 | . . . 4 | |
4 | 0re 7119 | . . . . 5 | |
5 | ltle 7198 | . . . . 5 | |
6 | 4, 5 | mpan 414 | . . . 4 |
7 | 2, 3, 6 | sylc 61 | . . 3 |
8 | 0le0 8128 | . . . 4 | |
9 | breq2 3789 | . . . 4 | |
10 | 8, 9 | mpbiri 166 | . . 3 |
11 | 7, 10 | jaoi 668 | . 2 |
12 | 1, 11 | sylbi 119 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 661 wceq 1284 wcel 1433 class class class wbr 3785 cr 6980 cc0 6981 clt 7153 cle 7154 cn 8039 cn0 8288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-iota 4887 df-fv 4930 df-ov 5535 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-inn 8040 df-n0 8289 |
This theorem is referenced by: nn0nlt0 8314 nn0ge0i 8315 nn0le0eq0 8316 nn0p1gt0 8317 0mnnnnn0 8320 nn0addge1 8334 nn0addge2 8335 nn0ge0d 8344 elnn0z 8364 nn0lt10b 8428 nn0ge0div 8434 nn0pnfge0 8866 0elfz 9132 fz0fzelfz0 9138 fz0fzdiffz0 9141 fzctr 9144 difelfzle 9145 elfzodifsumelfzo 9210 fvinim0ffz 9250 subfzo0 9251 adddivflid 9294 modqmuladdnn0 9370 modfzo0difsn 9397 bernneq 9593 bernneq3 9595 faclbnd 9668 faclbnd6 9671 facubnd 9672 ibcval5 9690 dvdseq 10248 evennn02n 10282 nn0ehalf 10303 nn0oddm1d2 10309 gcdn0gt0 10369 nn0gcdid0 10372 absmulgcd 10406 algcvgblem 10431 ialgcvga 10433 lcmgcdnn 10464 |
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