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Mirrors > Home > ILE Home > Th. List > elznn0nn | Unicode version |
Description: Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.) |
Ref | Expression |
---|---|
elznn0nn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 9269 |
. 2
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2 | andi 819 |
. . 3
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3 | df-3or 980 |
. . . 4
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4 | 3 | anbi2i 457 |
. . 3
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5 | nn0re 9199 |
. . . . . 6
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6 | 5 | pm4.71ri 392 |
. . . . 5
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7 | elnn0 9192 |
. . . . . . 7
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8 | orcom 729 |
. . . . . . 7
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9 | 7, 8 | bitri 184 |
. . . . . 6
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10 | 9 | anbi2i 457 |
. . . . 5
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11 | 6, 10 | bitri 184 |
. . . 4
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12 | 11 | orbi1i 764 |
. . 3
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13 | 2, 4, 12 | 3bitr4i 212 |
. 2
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14 | 1, 13 | bitri 184 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-sep 4133 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-i2m1 7930 ax-rnegex 7934 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-iota 5190 df-fv 5236 df-ov 5891 df-neg 8145 df-inn 8934 df-n0 9191 df-z 9268 |
This theorem is referenced by: peano2z 9303 zindd 9385 expcl2lemap 10546 mulexpzap 10574 expaddzap 10578 expmulzap 10580 absexpzap 11103 pcid 12337 mulgsubcl 13029 mulgneg 13033 |
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