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Mirrors > Home > ILE Home > Th. List > nnge1 | Unicode version |
Description: A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
nnge1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3941 |
. 2
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2 | breq2 3941 |
. 2
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3 | breq2 3941 |
. 2
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4 | breq2 3941 |
. 2
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5 | 1le1 8358 |
. 2
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6 | nnre 8751 |
. . 3
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7 | recn 7777 |
. . . . . 6
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8 | 7 | addid1d 7935 |
. . . . 5
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9 | 8 | breq2d 3949 |
. . . 4
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10 | 0lt1 7913 |
. . . . . . . 8
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11 | 0re 7790 |
. . . . . . . . 9
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12 | 1re 7789 |
. . . . . . . . 9
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13 | axltadd 7858 |
. . . . . . . . 9
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14 | 11, 12, 13 | mp3an12 1306 |
. . . . . . . 8
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15 | 10, 14 | mpi 15 |
. . . . . . 7
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16 | readdcl 7770 |
. . . . . . . . 9
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17 | 11, 16 | mpan2 422 |
. . . . . . . 8
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18 | peano2re 7922 |
. . . . . . . 8
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19 | lttr 7862 |
. . . . . . . . 9
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20 | 12, 19 | mp3an3 1305 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 17, 18, 20 | syl2anc 409 |
. . . . . . 7
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22 | 15, 21 | mpand 426 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | con3d 621 |
. . . . 5
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24 | lenlt 7864 |
. . . . . 6
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25 | 12, 17, 24 | sylancr 411 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | lenlt 7864 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 12, 18, 26 | sylancr 411 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 23, 25, 27 | 3imtr4d 202 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 9, 28 | sylbird 169 |
. . 3
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30 | 6, 29 | syl 14 |
. 2
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31 | 1, 2, 3, 4, 5, 30 | nnind 8760 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-pre-ltirr 7756 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-iota 5096 df-fv 5139 df-ov 5785 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-inn 8745 |
This theorem is referenced by: nnle1eq1 8768 nngt0 8769 nnnlt1 8770 nnrecgt0 8782 nnge1d 8787 elnnnn0c 9046 elnnz1 9101 zltp1le 9132 nn0ledivnn 9584 elfz1b 9901 fzo1fzo0n0 9991 elfzom1elp1fzo 10010 fzo0sn0fzo1 10029 nnlesq 10427 faclbnd 10519 faclbnd3 10521 cvgratz 11333 coprmgcdb 11805 isprm3 11835 pw2dvds 11880 oddennn 11941 |
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