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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 6407 | . . 3 | |
2 | 0npi 6411 | . . . . . 6 | |
3 | eleq1 2100 | . . . . . 6 | |
4 | 2, 3 | mtbiri 600 | . . . . 5 |
5 | 4 | con2i 557 | . . . 4 |
6 | 0elnn 4340 | . . . . . 6 | |
7 | 1, 6 | syl 14 | . . . . 5 |
8 | 7 | ord 643 | . . . 4 |
9 | 5, 8 | mpd 13 | . . 3 |
10 | 1, 9 | jca 290 | . 2 |
11 | nndceq0 4339 | . . . . . 6 DECID | |
12 | df-dc 743 | . . . . . 6 DECID | |
13 | 11, 12 | sylib 127 | . . . . 5 |
14 | 13 | anim1i 323 | . . . 4 |
15 | ancom 253 | . . . . 5 | |
16 | andi 731 | . . . . 5 | |
17 | 15, 16 | bitr3i 175 | . . . 4 |
18 | 14, 17 | sylib 127 | . . 3 |
19 | noel 3228 | . . . . . . . . 9 | |
20 | eleq2 2101 | . . . . . . . . 9 | |
21 | 19, 20 | mtbiri 600 | . . . . . . . 8 |
22 | 21 | pm2.21d 549 | . . . . . . 7 |
23 | 22 | impcom 116 | . . . . . 6 |
24 | 23 | a1i 9 | . . . . 5 |
25 | df-ne 2206 | . . . . . . 7 | |
26 | elni 6406 | . . . . . . . 8 | |
27 | 26 | simplbi2 367 | . . . . . . 7 |
28 | 25, 27 | syl5bir 142 | . . . . . 6 |
29 | 28 | adantld 263 | . . . . 5 |
30 | 24, 29 | jaod 637 | . . . 4 |
31 | 30 | adantr 261 | . . 3 |
32 | 18, 31 | mpd 13 | . 2 |
33 | 10, 32 | impbii 117 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 DECID wdc 742 wceq 1243 wcel 1393 wne 2204 c0 3224 com 4313 cnpi 6370 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 df-ni 6402 |
This theorem is referenced by: addclpi 6425 mulclpi 6426 mulcanpig 6433 addnidpig 6434 ltexpi 6435 ltmpig 6437 nnppipi 6441 archnqq 6515 enq0tr 6532 |
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