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Mirrors > Home > ILE Home > Th. List > nngt0 | Unicode version |
Description: A positive integer is positive. (Contributed by NM, 26-Sep-1999.) |
Ref | Expression |
---|---|
nngt0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 8695 | . 2 | |
2 | nnge1 8711 | . 2 | |
3 | 0lt1 7857 | . . 3 | |
4 | 0re 7734 | . . . 4 | |
5 | 1re 7733 | . . . 4 | |
6 | ltletr 7821 | . . . 4 | |
7 | 4, 5, 6 | mp3an12 1290 | . . 3 |
8 | 3, 7 | mpani 426 | . 2 |
9 | 1, 2, 8 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 class class class wbr 3899 cr 7587 cc0 7588 c1 7589 clt 7768 cle 7769 cn 8688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-iota 5058 df-fv 5101 df-ov 5745 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-inn 8689 |
This theorem is referenced by: nnap0 8717 nngt0i 8718 nn2ge 8721 nn1gt1 8722 nnsub 8727 nngt0d 8732 nnrecl 8943 nn0ge0 8970 0mnnnnn0 8977 elnnnn0b 8989 elnnz 9032 elnn0z 9035 ztri3or0 9064 nnm1ge0 9105 gtndiv 9114 nnrp 9419 nnledivrp 9521 fzo1fzo0n0 9928 ubmelfzo 9945 adddivflid 10033 flltdivnn0lt 10045 intfracq 10061 zmodcl 10085 zmodfz 10087 zmodid2 10093 m1modnnsub1 10111 expnnval 10264 nnlesq 10364 facdiv 10452 faclbnd 10455 bc0k 10470 dvdsval3 11424 nndivdvds 11426 moddvds 11429 evennn2n 11507 nnoddm1d2 11534 divalglemnn 11542 ndvdssub 11554 ndvdsadd 11555 modgcd 11606 sqgcd 11644 lcmgcdlem 11685 qredeu 11705 divdenle 11802 hashgcdlem 11830 znnen 11838 exmidunben 11866 |
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