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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 9204 | . . . 4 | |
2 | 1cnd 7923 | . . . 4 | |
3 | 1, 2 | negsubdid 8232 | . . 3 |
4 | znegcl 9230 | . . . 4 | |
5 | peano2z 9235 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | 3, 6 | eqeltrd 2247 | . 2 |
8 | 1, 2 | subcld 8217 | . . 3 |
9 | znegclb 9232 | . . 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 2141 (class class class)co 5850 cc 7759 c1 7762 caddc 7764 cmin 8077 cneg 8078 cz 9199 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-n0 9123 df-z 9200 |
This theorem is referenced by: zaddcllemneg 9238 zlem1lt 9255 zltlem1 9256 zextlt 9291 zeo 9304 eluzp1m1 9497 fz01en 9996 fzsuc2 10022 elfzm11 10034 uzdisj 10036 fzof 10087 fzoval 10091 elfzo 10092 fzodcel 10095 fzon 10109 fzoss2 10115 fzossrbm1 10116 fzosplitsnm1 10152 ubmelm1fzo 10169 elfzom1b 10172 fzosplitprm1 10177 fzoshftral 10181 fzofig 10375 uzsinds 10385 ser3mono 10421 iseqf1olemqcl 10429 iseqf1olemnab 10431 iseqf1olemab 10432 seq3f1olemqsumkj 10441 seq3f1olemqsum 10443 bcm1k 10681 bcn2 10685 bcp1m1 10686 bcpasc 10687 bccl 10688 zfz1isolemiso 10761 seq3coll 10764 resqrexlemcalc3 10967 resqrexlemnm 10969 fsumm1 11366 binomlem 11433 binom1dif 11437 isumsplit 11441 arisum2 11449 pwm1geoserap1 11458 mertenslemi1 11485 fprodm1 11548 fprodeq0 11567 zeo3 11814 oddm1even 11821 oddp1even 11822 zob 11837 nno 11852 isprm3 12059 prmdc 12071 isprm5 12083 phibnd 12158 hashdvds 12162 odzcllem 12183 odzdvds 12186 fldivp1 12287 pockthlem 12295 oddennn 12334 lgslem1 13654 lgsval2lem 13664 2sqlem8 13712 |
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