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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 8356 | . . . 4 | |
2 | 1cnd 7135 | . . . 4 | |
3 | 1, 2 | negsubdid 7434 | . . 3 |
4 | znegcl 8382 | . . . 4 | |
5 | peano2z 8387 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | 3, 6 | eqeltrd 2155 | . 2 |
8 | 1, 2 | subcld 7419 | . . 3 |
9 | znegclb 8384 | . . 3 | |
10 | 8, 9 | syl 14 | . 2 |
11 | 7, 10 | mpbird 165 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 103 wcel 1433 (class class class)co 5532 cc 6979 c1 6982 caddc 6984 cmin 7279 cneg 7280 cz 8351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 |
This theorem is referenced by: zaddcllemneg 8390 zlem1lt 8407 zltlem1 8408 zextlt 8439 zeo 8452 eluzp1m1 8642 fz01en 9072 fzsuc2 9096 elfzm11 9108 uzdisj 9110 fzof 9154 fzoval 9158 elfzo 9159 fzon 9175 fzoss2 9181 fzossrbm1 9182 fzosplitsnm1 9218 ubmelm1fzo 9235 elfzom1b 9238 fzosplitprm1 9243 fzoshftral 9247 fzofig 9424 uzsinds 9428 isermono 9457 bcm1k 9687 bcn2 9691 bcp1m1 9692 bcpasc 9693 bccl 9694 resqrexlemcalc3 9902 resqrexlemnm 9904 zeo3 10267 oddm1even 10274 oddp1even 10275 zob 10291 nno 10306 isprm3 10500 |
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