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Mirrors > Home > MPE Home > Th. List > zlem1lt | Structured version Visualization version Unicode version |
Description: Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
Ref | Expression |
---|---|
zlem1lt |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2zm 10977 |
. . 3
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2 | zltp1le 10983 |
. . 3
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3 | 1, 2 | sylan 474 |
. 2
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4 | zcn 10939 |
. . . . 5
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5 | ax-1cn 9594 |
. . . . 5
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6 | npcan 9881 |
. . . . 5
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7 | 4, 5, 6 | sylancl 667 |
. . . 4
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8 | 7 | adantr 467 |
. . 3
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9 | 8 | breq1d 4411 |
. 2
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10 | 3, 9 | bitr2d 258 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-resscn 9593 ax-1cn 9594 ax-icn 9595 ax-addcl 9596 ax-addrcl 9597 ax-mulcl 9598 ax-mulrcl 9599 ax-mulcom 9600 ax-addass 9601 ax-mulass 9602 ax-distr 9603 ax-i2m1 9604 ax-1ne0 9605 ax-1rid 9606 ax-rnegex 9607 ax-rrecex 9608 ax-cnre 9609 ax-pre-lttri 9610 ax-pre-lttrn 9611 ax-pre-ltadd 9612 ax-pre-mulgt0 9613 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-om 6690 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-sub 9859 df-neg 9860 df-nn 10607 df-n0 10867 df-z 10935 |
This theorem is referenced by: nn0lem1lt 10998 nnlem1lt 10999 zbtwnre 11259 uzdisj 11864 nn0disj 11904 fzon 11936 ssfzo12 12001 ceile 12073 cshwidxn 12905 bitsfzolem 14400 bitsfzolemOLD 14401 bitscmp 14405 bitsinv1lem 14408 hashdvds 14716 logf1o2 23588 ang180lem3 23733 lgsquadlem1 24275 frgrawopreglem2 25766 fzsplit3 28363 ballotlemfc0 29318 ballotlemfcc 29319 ballotlemimin 29331 ballotlemfrceq 29354 ballotlemfrcn0 29355 ballotlemiminOLD 29369 ballotlemfrceqOLD 29392 ballotlemfrcn0OLD 29393 poimirlem23 31956 poimirlem24 31957 irrapxlem3 35662 hashnzfz2 36664 fzdifsuc2 37524 stoweidlem26 37880 fourierdlem12 37975 nnsum3primesle9 38883 evengpop3 38887 fzoopth 39050 zgtp1leeq 40306 m1modmmod 40311 nnolog2flm1 40388 |
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