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Mirrors > Home > ILE Home > Th. List > peano2zm | Unicode version |
Description: "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.) |
Ref | Expression |
---|---|
peano2zm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9027 | . . . 4 | |
2 | 1cnd 7750 | . . . 4 | |
3 | 1, 2 | negsubdid 8056 | . . 3 |
4 | znegcl 9053 | . . . 4 | |
5 | peano2z 9058 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | 3, 6 | eqeltrd 2194 | . 2 |
8 | 1, 2 | subcld 8041 | . . 3 |
9 | znegclb 9055 | . . 3 | |
10 | 8, 9 | syl 14 | . 2 |
11 | 7, 10 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wcel 1465 (class class class)co 5742 cc 7586 c1 7589 caddc 7591 cmin 7901 cneg 7902 cz 9022 |
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-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 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-3or 948 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-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 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-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 |
This theorem is referenced by: zaddcllemneg 9061 zlem1lt 9078 zltlem1 9079 zextlt 9111 zeo 9124 eluzp1m1 9317 fz01en 9801 fzsuc2 9827 elfzm11 9839 uzdisj 9841 fzof 9889 fzoval 9893 elfzo 9894 fzodcel 9897 fzon 9911 fzoss2 9917 fzossrbm1 9918 fzosplitsnm1 9954 ubmelm1fzo 9971 elfzom1b 9974 fzosplitprm1 9979 fzoshftral 9983 fzofig 10173 uzsinds 10183 ser3mono 10219 iseqf1olemqcl 10227 iseqf1olemnab 10229 iseqf1olemab 10230 seq3f1olemqsumkj 10239 seq3f1olemqsum 10241 bcm1k 10474 bcn2 10478 bcp1m1 10479 bcpasc 10480 bccl 10481 zfz1isolemiso 10550 seq3coll 10553 resqrexlemcalc3 10756 resqrexlemnm 10758 fsumm1 11153 binomlem 11220 binom1dif 11224 isumsplit 11228 arisum2 11236 pwm1geoserap1 11245 mertenslemi1 11272 zeo3 11492 oddm1even 11499 oddp1even 11500 zob 11515 nno 11530 isprm3 11726 phibnd 11820 hashdvds 11824 oddennn 11832 |
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