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Mirrors > Home > ILE Home > Th. List > elz2 | Unicode version |
Description: Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the positive integers. (Contributed by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
elz2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elznn0 9037 | . 2 | |
2 | nn0p1nn 8984 | . . . . . 6 | |
3 | 2 | adantl 275 | . . . . 5 |
4 | 1nn 8699 | . . . . . 6 | |
5 | 4 | a1i 9 | . . . . 5 |
6 | recn 7721 | . . . . . . . 8 | |
7 | 6 | adantr 274 | . . . . . . 7 |
8 | ax-1cn 7681 | . . . . . . 7 | |
9 | pncan 7936 | . . . . . . 7 | |
10 | 7, 8, 9 | sylancl 409 | . . . . . 6 |
11 | 10 | eqcomd 2123 | . . . . 5 |
12 | rspceov 5781 | . . . . 5 | |
13 | 3, 5, 11, 12 | syl3anc 1201 | . . . 4 |
14 | 4 | a1i 9 | . . . . 5 |
15 | 6 | adantr 274 | . . . . . . 7 |
16 | negsub 7978 | . . . . . . 7 | |
17 | 8, 15, 16 | sylancr 410 | . . . . . 6 |
18 | simpr 109 | . . . . . . 7 | |
19 | nnnn0addcl 8975 | . . . . . . 7 | |
20 | 4, 18, 19 | sylancr 410 | . . . . . 6 |
21 | 17, 20 | eqeltrrd 2195 | . . . . 5 |
22 | nncan 7959 | . . . . . . 7 | |
23 | 8, 15, 22 | sylancr 410 | . . . . . 6 |
24 | 23 | eqcomd 2123 | . . . . 5 |
25 | rspceov 5781 | . . . . 5 | |
26 | 14, 21, 24, 25 | syl3anc 1201 | . . . 4 |
27 | 13, 26 | jaodan 771 | . . 3 |
28 | nnre 8695 | . . . . . . 7 | |
29 | nnre 8695 | . . . . . . 7 | |
30 | resubcl 7994 | . . . . . . 7 | |
31 | 28, 29, 30 | syl2an 287 | . . . . . 6 |
32 | nnz 9041 | . . . . . . . 8 | |
33 | nnz 9041 | . . . . . . . 8 | |
34 | zletric 9066 | . . . . . . . 8 | |
35 | 32, 33, 34 | syl2anr 288 | . . . . . . 7 |
36 | nnnn0 8952 | . . . . . . . . 9 | |
37 | nnnn0 8952 | . . . . . . . . 9 | |
38 | nn0sub 9088 | . . . . . . . . 9 | |
39 | 36, 37, 38 | syl2anr 288 | . . . . . . . 8 |
40 | nn0sub 9088 | . . . . . . . . . 10 | |
41 | 37, 36, 40 | syl2an 287 | . . . . . . . . 9 |
42 | nncn 8696 | . . . . . . . . . . 11 | |
43 | nncn 8696 | . . . . . . . . . . 11 | |
44 | negsubdi2 7989 | . . . . . . . . . . 11 | |
45 | 42, 43, 44 | syl2an 287 | . . . . . . . . . 10 |
46 | 45 | eleq1d 2186 | . . . . . . . . 9 |
47 | 41, 46 | bitr4d 190 | . . . . . . . 8 |
48 | 39, 47 | orbi12d 767 | . . . . . . 7 |
49 | 35, 48 | mpbid 146 | . . . . . 6 |
50 | 31, 49 | jca 304 | . . . . 5 |
51 | eleq1 2180 | . . . . . 6 | |
52 | eleq1 2180 | . . . . . . 7 | |
53 | negeq 7923 | . . . . . . . 8 | |
54 | 53 | eleq1d 2186 | . . . . . . 7 |
55 | 52, 54 | orbi12d 767 | . . . . . 6 |
56 | 51, 55 | anbi12d 464 | . . . . 5 |
57 | 50, 56 | syl5ibrcom 156 | . . . 4 |
58 | 57 | rexlimivv 2532 | . . 3 |
59 | 27, 58 | impbii 125 | . 2 |
60 | 1, 59 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wo 682 wceq 1316 wcel 1465 wrex 2394 class class class wbr 3899 (class class class)co 5742 cc 7586 cr 7587 c1 7589 caddc 7591 cle 7769 cmin 7901 cneg 7902 cn 8688 cn0 8945 cz 9022 |
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-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-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 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 8689 df-n0 8946 df-z 9023 |
This theorem is referenced by: dfz2 9091 |
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