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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 9187 | . . . 4 | |
2 | 1cnd 7906 | . . . 4 | |
3 | 1, 2 | negsubdid 8215 | . . 3 |
4 | znegcl 9213 | . . . 4 | |
5 | peano2z 9218 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | 3, 6 | eqeltrd 2241 | . 2 |
8 | 1, 2 | subcld 8200 | . . 3 |
9 | znegclb 9215 | . . 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 2135 (class class class)co 5836 cc 7742 c1 7745 caddc 7747 cmin 8060 cneg 8061 cz 9182 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 |
This theorem is referenced by: zaddcllemneg 9221 zlem1lt 9238 zltlem1 9239 zextlt 9274 zeo 9287 eluzp1m1 9480 fz01en 9978 fzsuc2 10004 elfzm11 10016 uzdisj 10018 fzof 10069 fzoval 10073 elfzo 10074 fzodcel 10077 fzon 10091 fzoss2 10097 fzossrbm1 10098 fzosplitsnm1 10134 ubmelm1fzo 10151 elfzom1b 10154 fzosplitprm1 10159 fzoshftral 10163 fzofig 10357 uzsinds 10367 ser3mono 10403 iseqf1olemqcl 10411 iseqf1olemnab 10413 iseqf1olemab 10414 seq3f1olemqsumkj 10423 seq3f1olemqsum 10425 bcm1k 10662 bcn2 10666 bcp1m1 10667 bcpasc 10668 bccl 10669 zfz1isolemiso 10738 seq3coll 10741 resqrexlemcalc3 10944 resqrexlemnm 10946 fsumm1 11343 binomlem 11410 binom1dif 11414 isumsplit 11418 arisum2 11426 pwm1geoserap1 11435 mertenslemi1 11462 fprodm1 11525 fprodeq0 11544 zeo3 11790 oddm1even 11797 oddp1even 11798 zob 11813 nno 11828 isprm3 12029 prmdc 12041 isprm5 12053 phibnd 12128 hashdvds 12132 odzcllem 12153 odzdvds 12156 fldivp1 12257 pockthlem 12265 oddennn 12268 |
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