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Mirrors > Home > ILE Home > Th. List > zltp1le | Unicode version |
Description: Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
zltp1le |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1 8337 |
. . . 4
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2 | 1 | a1i 9 |
. . 3
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3 | znnsub 8695 |
. . 3
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4 | zre 8648 |
. . . 4
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5 | zre 8648 |
. . . 4
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6 | 1re 7388 |
. . . . 5
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7 | leaddsub2 7818 |
. . . . 5
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8 | 6, 7 | mp3an2 1257 |
. . . 4
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9 | 4, 5, 8 | syl2an 283 |
. . 3
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10 | 2, 3, 9 | 3imtr4d 201 |
. 2
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11 | 4 | adantr 270 |
. . . 4
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12 | 11 | ltp1d 8283 |
. . 3
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13 | peano2re 7519 |
. . . . 5
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14 | 11, 13 | syl 14 |
. . . 4
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15 | 5 | adantl 271 |
. . . 4
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16 | ltletr 7475 |
. . . 4
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17 | 11, 14, 15, 16 | syl3anc 1170 |
. . 3
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18 | 12, 17 | mpand 420 |
. 2
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19 | 10, 18 | impbid 127 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-addcom 7346 ax-addass 7348 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-0id 7354 ax-rnegex 7355 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-ltadd 7362 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-id 4083 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-iota 4932 df-fun 4969 df-fv 4975 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-inn 8315 df-n0 8564 df-z 8645 |
This theorem is referenced by: zleltp1 8699 zlem1lt 8700 zgt0ge1 8702 nnltp1le 8704 nn0ltp1le 8706 btwnnz 8734 uzind2 8752 fzind 8755 eluzp1l 8936 eluz2b1 8981 zltaddlt1le 9316 fzsplit2 9357 m1modge3gt1 9665 ibcval5 10004 nn0o1gt2 10683 divalglemnqt 10698 zsupcllemstep 10719 infssuzex 10723 isprm3 10878 dvdsnprmd 10885 prmgt1 10891 oddprmge3 10894 znege1 10934 hashdvds 10975 |
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