Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7085 | . . 3 | |
2 | 0npi 7089 | . . . . . 6 | |
3 | eleq1 2180 | . . . . . 6 | |
4 | 2, 3 | mtbiri 649 | . . . . 5 |
5 | 4 | con2i 601 | . . . 4 |
6 | 0elnn 4502 | . . . . . 6 | |
7 | 1, 6 | syl 14 | . . . . 5 |
8 | 7 | ord 698 | . . . 4 |
9 | 5, 8 | mpd 13 | . . 3 |
10 | 1, 9 | jca 304 | . 2 |
11 | nndceq0 4501 | . . . . . 6 DECID | |
12 | df-dc 805 | . . . . . 6 DECID | |
13 | 11, 12 | sylib 121 | . . . . 5 |
14 | 13 | anim1i 338 | . . . 4 |
15 | ancom 264 | . . . . 5 | |
16 | andi 792 | . . . . 5 | |
17 | 15, 16 | bitr3i 185 | . . . 4 |
18 | 14, 17 | sylib 121 | . . 3 |
19 | noel 3337 | . . . . . . . . 9 | |
20 | eleq2 2181 | . . . . . . . . 9 | |
21 | 19, 20 | mtbiri 649 | . . . . . . . 8 |
22 | 21 | pm2.21d 593 | . . . . . . 7 |
23 | 22 | impcom 124 | . . . . . 6 |
24 | 23 | a1i 9 | . . . . 5 |
25 | df-ne 2286 | . . . . . . 7 | |
26 | elni 7084 | . . . . . . . 8 | |
27 | 26 | simplbi2 382 | . . . . . . 7 |
28 | 25, 27 | syl5bir 152 | . . . . . 6 |
29 | 28 | adantld 276 | . . . . 5 |
30 | 24, 29 | jaod 691 | . . . 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 682 DECID wdc 804 wceq 1316 wcel 1465 wne 2285 c0 3333 com 4474 cnpi 7048 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-suc 4263 df-iom 4475 df-ni 7080 |
This theorem is referenced by: addclpi 7103 mulclpi 7104 mulcanpig 7111 addnidpig 7112 ltexpi 7113 ltmpig 7115 nnppipi 7119 archnqq 7193 enq0tr 7210 |
Copyright terms: Public domain | W3C validator |