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Mirrors > Home > ILE Home > Th. List > fz0fzelfz0 | Unicode version |
Description: If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.) |
Ref | Expression |
---|---|
fz0fzelfz0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz2nn0 9860 | . . . 4 | |
2 | elfz2 9765 | . . . . . 6 | |
3 | simplr 504 | . . . . . . . . . . . . . . . . 17 | |
4 | 0red 7735 | . . . . . . . . . . . . . . . . . . . 20 | |
5 | nn0re 8954 | . . . . . . . . . . . . . . . . . . . . 21 | |
6 | 5 | adantr 274 | . . . . . . . . . . . . . . . . . . . 20 |
7 | zre 9026 | . . . . . . . . . . . . . . . . . . . . 21 | |
8 | 7 | adantl 275 | . . . . . . . . . . . . . . . . . . . 20 |
9 | 4, 6, 8 | 3jca 1146 | . . . . . . . . . . . . . . . . . . 19 |
10 | 9 | adantr 274 | . . . . . . . . . . . . . . . . . 18 |
11 | nn0ge0 8970 | . . . . . . . . . . . . . . . . . . . 20 | |
12 | 11 | adantr 274 | . . . . . . . . . . . . . . . . . . 19 |
13 | 12 | anim1i 338 | . . . . . . . . . . . . . . . . . 18 |
14 | letr 7815 | . . . . . . . . . . . . . . . . . 18 | |
15 | 10, 13, 14 | sylc 62 | . . . . . . . . . . . . . . . . 17 |
16 | elnn0z 9035 | . . . . . . . . . . . . . . . . 17 | |
17 | 3, 15, 16 | sylanbrc 413 | . . . . . . . . . . . . . . . 16 |
18 | 17 | exp31 361 | . . . . . . . . . . . . . . 15 |
19 | 18 | com23 78 | . . . . . . . . . . . . . 14 |
20 | 19 | 3ad2ant1 987 | . . . . . . . . . . . . 13 |
21 | 20 | com13 80 | . . . . . . . . . . . 12 |
22 | 21 | adantrd 277 | . . . . . . . . . . 11 |
23 | 22 | 3ad2ant3 989 | . . . . . . . . . 10 |
24 | 23 | imp 123 | . . . . . . . . 9 |
25 | 24 | imp 123 | . . . . . . . 8 |
26 | simpr2 973 | . . . . . . . 8 | |
27 | simplrr 510 | . . . . . . . 8 | |
28 | 25, 26, 27 | 3jca 1146 | . . . . . . 7 |
29 | 28 | ex 114 | . . . . . 6 |
30 | 2, 29 | sylbi 120 | . . . . 5 |
31 | 30 | com12 30 | . . . 4 |
32 | 1, 31 | sylbi 120 | . . 3 |
33 | 32 | imp 123 | . 2 |
34 | elfz2nn0 9860 | . 2 | |
35 | 33, 34 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wcel 1465 class class class wbr 3899 (class class class)co 5742 cr 7587 cc0 7588 cle 7769 cn0 8945 cz 9022 cfz 9758 |
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-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 8689 df-n0 8946 df-z 9023 df-uz 9295 df-fz 9759 |
This theorem is referenced by: fz0fzdiffz0 9875 |
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