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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 8647 |
. . . 4
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2 | 1 | a1i 9 |
. . 3
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3 | znnsub 9003 |
. . 3
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4 | zre 8956 |
. . . 4
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5 | zre 8956 |
. . . 4
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6 | 1re 7683 |
. . . . 5
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7 | leaddsub2 8114 |
. . . . 5
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8 | 6, 7 | mp3an2 1284 |
. . . 4
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9 | 4, 5, 8 | syl2an 285 |
. . 3
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10 | 2, 3, 9 | 3imtr4d 202 |
. 2
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11 | 4 | adantr 272 |
. . . 4
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12 | 11 | ltp1d 8592 |
. . 3
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13 | peano2re 7815 |
. . . . 5
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14 | 11, 13 | syl 14 |
. . . 4
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15 | 5 | adantl 273 |
. . . 4
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16 | ltletr 7770 |
. . . 4
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17 | 11, 14, 15, 16 | syl3anc 1197 |
. . 3
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18 | 12, 17 | mpand 423 |
. 2
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19 | 10, 18 | impbid 128 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-ltadd 7655 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-iota 5044 df-fun 5081 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-inn 8625 df-n0 8876 df-z 8953 |
This theorem is referenced by: zleltp1 9007 zlem1lt 9008 zgt0ge1 9010 nnltp1le 9012 nn0ltp1le 9014 btwnnz 9043 uzind2 9061 fzind 9064 btwnapz 9079 eluzp1l 9246 eluz2b1 9291 zltaddlt1le 9676 fzsplit2 9717 m1modge3gt1 10031 seq3f1olemqsumkj 10158 seq3f1olemqsumk 10159 bcval5 10396 seq3coll 10472 cvgratnnlemseq 11181 nn0o1gt2 11444 divalglemnqt 11459 zsupcllemstep 11480 infssuzex 11484 isprm3 11639 dvdsnprmd 11646 prmgt1 11652 oddprmge3 11655 znege1 11695 hashdvds 11736 |
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