Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 9892 | . . . 4 | |
2 | elfz2 9797 | . . . . . 6 | |
3 | nn0re 8986 | . . . . . . . . . . . . . . . . . 18 | |
4 | nn0re 8986 | . . . . . . . . . . . . . . . . . 18 | |
5 | zre 9058 | . . . . . . . . . . . . . . . . . 18 | |
6 | 3, 4, 5 | 3anim123i 1166 | . . . . . . . . . . . . . . . . 17 |
7 | 6 | 3expa 1181 | . . . . . . . . . . . . . . . 16 |
8 | letr 7847 | . . . . . . . . . . . . . . . 16 | |
9 | 7, 8 | syl 14 | . . . . . . . . . . . . . . 15 |
10 | simplll 522 | . . . . . . . . . . . . . . . . 17 | |
11 | simpr 109 | . . . . . . . . . . . . . . . . . . 19 | |
12 | 11 | adantr 274 | . . . . . . . . . . . . . . . . . 18 |
13 | elnn0z 9067 | . . . . . . . . . . . . . . . . . . . . . 22 | |
14 | 0red 7767 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 | |
15 | zre 9058 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27 | |
16 | 15 | adantr 274 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 |
17 | 5 | adantl 275 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 |
18 | letr 7847 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 | |
19 | 14, 16, 17, 18 | syl3anc 1216 | . . . . . . . . . . . . . . . . . . . . . . . . 25 |
20 | 19 | exp4b 364 | . . . . . . . . . . . . . . . . . . . . . . . 24 |
21 | 20 | com23 78 | . . . . . . . . . . . . . . . . . . . . . . 23 |
22 | 21 | imp 123 | . . . . . . . . . . . . . . . . . . . . . 22 |
23 | 13, 22 | sylbi 120 | . . . . . . . . . . . . . . . . . . . . 21 |
24 | 23 | adantr 274 | . . . . . . . . . . . . . . . . . . . 20 |
25 | 24 | imp 123 | . . . . . . . . . . . . . . . . . . 19 |
26 | 25 | imp 123 | . . . . . . . . . . . . . . . . . 18 |
27 | elnn0z 9067 | . . . . . . . . . . . . . . . . . 18 | |
28 | 12, 26, 27 | sylanbrc 413 | . . . . . . . . . . . . . . . . 17 |
29 | simpr 109 | . . . . . . . . . . . . . . . . 17 | |
30 | 10, 28, 29 | 3jca 1161 | . . . . . . . . . . . . . . . 16 |
31 | 30 | ex 114 | . . . . . . . . . . . . . . 15 |
32 | 9, 31 | syld 45 | . . . . . . . . . . . . . 14 |
33 | 32 | exp4b 364 | . . . . . . . . . . . . 13 |
34 | 33 | com23 78 | . . . . . . . . . . . 12 |
35 | 34 | 3impia 1178 | . . . . . . . . . . 11 |
36 | 35 | com13 80 | . . . . . . . . . 10 |
37 | 36 | adantr 274 | . . . . . . . . 9 |
38 | 37 | com12 30 | . . . . . . . 8 |
39 | 38 | 3ad2ant3 1004 | . . . . . . 7 |
40 | 39 | imp 123 | . . . . . 6 |
41 | 2, 40 | sylbi 120 | . . . . 5 |
42 | 41 | com12 30 | . . . 4 |
43 | 1, 42 | sylbi 120 | . . 3 |
44 | 43 | imp 123 | . 2 |
45 | elfz2nn0 9892 | . 2 | |
46 | 44, 45 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7619 cc0 7620 cle 7801 cn0 8977 cz 9054 cfz 9790 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |