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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 7241 | . . 3 | |
2 | 0npi 7245 | . . . . . 6 | |
3 | eleq1 2227 | . . . . . 6 | |
4 | 2, 3 | mtbiri 665 | . . . . 5 |
5 | 4 | con2i 617 | . . . 4 |
6 | 0elnn 4590 | . . . . . 6 | |
7 | 1, 6 | syl 14 | . . . . 5 |
8 | 7 | ord 714 | . . . 4 |
9 | 5, 8 | mpd 13 | . . 3 |
10 | 1, 9 | jca 304 | . 2 |
11 | nndceq0 4589 | . . . . . 6 DECID | |
12 | df-dc 825 | . . . . . 6 DECID | |
13 | 11, 12 | sylib 121 | . . . . 5 |
14 | 13 | anim1i 338 | . . . 4 |
15 | ancom 264 | . . . . 5 | |
16 | andi 808 | . . . . 5 | |
17 | 15, 16 | bitr3i 185 | . . . 4 |
18 | 14, 17 | sylib 121 | . . 3 |
19 | noel 3408 | . . . . . . . . 9 | |
20 | eleq2 2228 | . . . . . . . . 9 | |
21 | 19, 20 | mtbiri 665 | . . . . . . . 8 |
22 | 21 | pm2.21d 609 | . . . . . . 7 |
23 | 22 | impcom 124 | . . . . . 6 |
24 | 23 | a1i 9 | . . . . 5 |
25 | df-ne 2335 | . . . . . . 7 | |
26 | elni 7240 | . . . . . . . 8 | |
27 | 26 | simplbi2 383 | . . . . . . 7 |
28 | 25, 27 | syl5bir 152 | . . . . . 6 |
29 | 28 | adantld 276 | . . . . 5 |
30 | 24, 29 | jaod 707 | . . . 4 |
31 | 30 | adantr 274 | . . 3 |
32 | 18, 31 | mpd 13 | . 2 |
33 | 10, 32 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 824 wceq 1342 wcel 2135 wne 2334 c0 3404 com 4561 cnpi 7204 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-suc 4343 df-iom 4562 df-ni 7236 |
This theorem is referenced by: addclpi 7259 mulclpi 7260 mulcanpig 7267 addnidpig 7268 ltexpi 7269 ltmpig 7271 nnppipi 7275 archnqq 7349 enq0tr 7366 |
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