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Mirrors > Home > ILE Home > Th. List > elni2 | Unicode version |
Description: Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) |
Ref | Expression |
---|---|
elni2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7307 |
. . 3
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2 | 0npi 7311 |
. . . . . 6
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3 | eleq1 2240 |
. . . . . 6
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4 | 2, 3 | mtbiri 675 |
. . . . 5
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5 | 4 | con2i 627 |
. . . 4
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6 | 0elnn 4618 |
. . . . . 6
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7 | 1, 6 | syl 14 |
. . . . 5
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8 | 7 | ord 724 |
. . . 4
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9 | 5, 8 | mpd 13 |
. . 3
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10 | 1, 9 | jca 306 |
. 2
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11 | nndceq0 4617 |
. . . . . 6
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12 | df-dc 835 |
. . . . . 6
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13 | 11, 12 | sylib 122 |
. . . . 5
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14 | 13 | anim1i 340 |
. . . 4
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15 | ancom 266 |
. . . . 5
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16 | andi 818 |
. . . . 5
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17 | 15, 16 | bitr3i 186 |
. . . 4
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18 | 14, 17 | sylib 122 |
. . 3
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19 | noel 3426 |
. . . . . . . . 9
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20 | eleq2 2241 |
. . . . . . . . 9
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21 | 19, 20 | mtbiri 675 |
. . . . . . . 8
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22 | 21 | pm2.21d 619 |
. . . . . . 7
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23 | 22 | impcom 125 |
. . . . . 6
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24 | 23 | a1i 9 |
. . . . 5
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25 | df-ne 2348 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | elni 7306 |
. . . . . . . 8
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27 | 26 | simplbi2 385 |
. . . . . . 7
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28 | 25, 27 | biimtrrid 153 |
. . . . . 6
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29 | 28 | adantld 278 |
. . . . 5
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30 | 24, 29 | jaod 717 |
. . . 4
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31 | 30 | adantr 276 |
. . 3
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32 | 18, 31 | mpd 13 |
. 2
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33 | 10, 32 | impbii 126 |
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-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-uni 3810 df-int 3845 df-suc 4371 df-iom 4590 df-ni 7302 |
This theorem is referenced by: addclpi 7325 mulclpi 7326 mulcanpig 7333 addnidpig 7334 ltexpi 7335 ltmpig 7337 nnppipi 7341 archnqq 7415 enq0tr 7432 |
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