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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 9207 |
. 2
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2 | nnre 8955 |
. . . 4
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3 | nngt0 8973 |
. . . 4
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4 | 0re 7986 |
. . . . 5
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5 | ltle 8074 |
. . . . 5
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6 | 4, 5 | mpan 424 |
. . . 4
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7 | 2, 3, 6 | sylc 62 |
. . 3
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8 | 0le0 9037 |
. . . 4
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9 | breq2 4022 |
. . . 4
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10 | 8, 9 | mpbiri 168 |
. . 3
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11 | 7, 10 | jaoi 717 |
. 2
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12 | 1, 11 | sylbi 121 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-i2m1 7945 ax-0lt1 7946 ax-0id 7948 ax-rnegex 7949 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-ltadd 7956 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-iota 5196 df-fv 5243 df-ov 5898 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-inn 8949 df-n0 9206 |
This theorem is referenced by: nn0nlt0 9231 nn0ge0i 9232 nn0le0eq0 9233 nn0p1gt0 9234 0mnnnnn0 9237 nn0addge1 9251 nn0addge2 9252 nn0ge0d 9261 elnn0z 9295 nn0negleid 9350 nn0lt10b 9362 nn0ge0div 9369 nn0pnfge0 9820 xnn0xadd0 9896 0elfz 10147 fz0fzelfz0 10156 fz0fzdiffz0 10159 fzctr 10162 difelfzle 10163 elfzodifsumelfzo 10230 fvinim0ffz 10270 subfzo0 10271 adddivflid 10322 modqmuladdnn0 10398 modfzo0difsn 10425 uzennn 10466 bernneq 10671 bernneq3 10673 zzlesq 10719 faclbnd 10752 faclbnd6 10755 facubnd 10756 bcval5 10774 fihashneq0 10805 dvdseq 11885 evennn02n 11918 nn0ehalf 11939 nn0oddm1d2 11945 gcdn0gt0 12010 nn0gcdid0 12013 absmulgcd 12049 algcvgblem 12080 algcvga 12082 lcmgcdnn 12113 hashgcdlem 12269 odzdvds 12276 pcfaclem 12380 znnen 12448 logbgcd1irr 14837 lgsdinn0 14902 |
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