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Mirrors > Home > ILE Home > Th. List > elfz0fzfz0 | Unicode version |
Description: A member of a finite set of sequential nonnegative integers is a member of a finite set of sequential nonnegative integers with a member of a finite set of sequential nonnegative integers starting at the upper bound of the first interval. (Contributed by Alexander van der Vekens, 27-May-2018.) |
Ref | Expression |
---|---|
elfz0fzfz0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz2nn0 10082 | . . . 4 | |
2 | elfz2 9986 | . . . . . 6 | |
3 | nn0re 9158 | . . . . . . . . . . . . . . . . . 18 | |
4 | nn0re 9158 | . . . . . . . . . . . . . . . . . 18 | |
5 | zre 9230 | . . . . . . . . . . . . . . . . . 18 | |
6 | 3, 4, 5 | 3anim123i 1184 | . . . . . . . . . . . . . . . . 17 |
7 | 6 | 3expa 1203 | . . . . . . . . . . . . . . . 16 |
8 | letr 8014 | . . . . . . . . . . . . . . . 16 | |
9 | 7, 8 | syl 14 | . . . . . . . . . . . . . . 15 |
10 | simplll 533 | . . . . . . . . . . . . . . . . 17 | |
11 | simpr 110 | . . . . . . . . . . . . . . . . . . 19 | |
12 | 11 | adantr 276 | . . . . . . . . . . . . . . . . . 18 |
13 | elnn0z 9239 | . . . . . . . . . . . . . . . . . . . . . 22 | |
14 | 0red 7933 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 | |
15 | zre 9230 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27 | |
16 | 15 | adantr 276 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 |
17 | 5 | adantl 277 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 |
18 | letr 8014 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 | |
19 | 14, 16, 17, 18 | syl3anc 1238 | . . . . . . . . . . . . . . . . . . . . . . . . 25 |
20 | 19 | exp4b 367 | . . . . . . . . . . . . . . . . . . . . . . . 24 |
21 | 20 | com23 78 | . . . . . . . . . . . . . . . . . . . . . . 23 |
22 | 21 | imp 124 | . . . . . . . . . . . . . . . . . . . . . 22 |
23 | 13, 22 | sylbi 121 | . . . . . . . . . . . . . . . . . . . . 21 |
24 | 23 | adantr 276 | . . . . . . . . . . . . . . . . . . . 20 |
25 | 24 | imp 124 | . . . . . . . . . . . . . . . . . . 19 |
26 | 25 | imp 124 | . . . . . . . . . . . . . . . . . 18 |
27 | elnn0z 9239 | . . . . . . . . . . . . . . . . . 18 | |
28 | 12, 26, 27 | sylanbrc 417 | . . . . . . . . . . . . . . . . 17 |
29 | simpr 110 | . . . . . . . . . . . . . . . . 17 | |
30 | 10, 28, 29 | 3jca 1177 | . . . . . . . . . . . . . . . 16 |
31 | 30 | ex 115 | . . . . . . . . . . . . . . 15 |
32 | 9, 31 | syld 45 | . . . . . . . . . . . . . 14 |
33 | 32 | exp4b 367 | . . . . . . . . . . . . 13 |
34 | 33 | com23 78 | . . . . . . . . . . . 12 |
35 | 34 | 3impia 1200 | . . . . . . . . . . 11 |
36 | 35 | com13 80 | . . . . . . . . . 10 |
37 | 36 | adantr 276 | . . . . . . . . 9 |
38 | 37 | com12 30 | . . . . . . . 8 |
39 | 38 | 3ad2ant3 1020 | . . . . . . 7 |
40 | 39 | imp 124 | . . . . . 6 |
41 | 2, 40 | sylbi 121 | . . . . 5 |
42 | 41 | com12 30 | . . . 4 |
43 | 1, 42 | sylbi 121 | . . 3 |
44 | 43 | imp 124 | . 2 |
45 | elfz2nn0 10082 | . 2 | |
46 | 44, 45 | sylibr 134 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 w3a 978 wcel 2146 class class class wbr 3998 (class class class)co 5865 cr 7785 cc0 7786 cle 7967 cn0 9149 cz 9226 cfz 9979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-addcom 7886 ax-addass 7888 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-0id 7894 ax-rnegex 7895 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-ltadd 7902 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-inn 8893 df-n0 9150 df-z 9227 df-uz 9502 df-fz 9980 |
This theorem is referenced by: (None) |
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