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Mirrors > Home > ILE Home > Th. List > elznn0 | Unicode version |
Description: Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
elznn0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 8806 |
. 2
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2 | elnn0 8729 |
. . . . . 6
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3 | 2 | a1i 9 |
. . . . 5
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4 | elnn0 8729 |
. . . . . 6
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5 | recn 7529 |
. . . . . . . . 9
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6 | 0cn 7534 |
. . . . . . . . 9
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7 | negcon1 7788 |
. . . . . . . . 9
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8 | 5, 6, 7 | sylancl 405 |
. . . . . . . 8
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9 | neg0 7782 |
. . . . . . . . . 10
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10 | 9 | eqeq1i 2096 |
. . . . . . . . 9
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11 | eqcom 2091 |
. . . . . . . . 9
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12 | 10, 11 | bitri 183 |
. . . . . . . 8
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13 | 8, 12 | syl6bb 195 |
. . . . . . 7
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14 | 13 | orbi2d 740 |
. . . . . 6
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15 | 4, 14 | syl5bb 191 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 3, 15 | orbi12d 743 |
. . . 4
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17 | 3orass 928 |
. . . . 5
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18 | orcom 683 |
. . . . 5
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19 | orordir 727 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 17, 18, 19 | 3bitrri 206 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 16, 20 | syl6rbb 196 |
. . 3
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22 | 21 | pm5.32i 443 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 1, 22 | bitri 183 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-setind 4366 ax-resscn 7491 ax-1cn 7492 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-addcom 7499 ax-addass 7501 ax-distr 7503 ax-i2m1 7504 ax-0id 7507 ax-rnegex 7508 ax-cnre 7510 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-sub 7709 df-neg 7710 df-n0 8728 df-z 8805 |
This theorem is referenced by: peano2z 8840 zmulcl 8857 elz2 8872 expnegzap 10043 expaddzaplem 10052 odd2np1 11205 bezoutlemzz 11323 bezoutlemaz 11324 bezoutlembz 11325 |
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