Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elnn1uz2 | Unicode version |
Description: A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.) |
Ref | Expression |
---|---|
elnn1uz2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 685 | . . . 4 | |
2 | nnz 9031 | . . . . 5 | |
3 | 1z 9038 | . . . . . . . 8 | |
4 | zdceq 9084 | . . . . . . . 8 DECID | |
5 | 3, 4 | mpan2 421 | . . . . . . 7 DECID |
6 | df-dc 805 | . . . . . . 7 DECID | |
7 | 5, 6 | sylib 121 | . . . . . 6 |
8 | df-ne 2286 | . . . . . . 7 | |
9 | 8 | orbi2i 736 | . . . . . 6 |
10 | 7, 9 | sylibr 133 | . . . . 5 |
11 | 2, 10 | syl 14 | . . . 4 |
12 | ordi 790 | . . . 4 | |
13 | 1, 11, 12 | sylanbrc 413 | . . 3 |
14 | eluz2b3 9354 | . . . 4 | |
15 | 14 | orbi2i 736 | . . 3 |
16 | 13, 15 | sylibr 133 | . 2 |
17 | 1nn 8695 | . . . 4 | |
18 | eleq1 2180 | . . . 4 | |
19 | 17, 18 | mpbiri 167 | . . 3 |
20 | eluz2nn 9320 | . . 3 | |
21 | 19, 20 | jaoi 690 | . 2 |
22 | 16, 21 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 682 DECID wdc 804 wceq 1316 wcel 1465 wne 2285 cfv 5093 c1 7589 cn 8684 c2 8735 cz 9012 cuz 9282 |
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-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8685 df-2 8743 df-n0 8936 df-z 9013 df-uz 9283 |
This theorem is referenced by: indstr2 9359 dfphi2 11807 |
Copyright terms: Public domain | W3C validator |