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Mirrors > Home > ILE Home > Th. List > elfzonelfzo | Unicode version |
Description: If an element of a half-open integer range is not contained in the lower subrange, it must be in the upper subrange. (Contributed by Alexander van der Vekens, 30-Mar-2018.) |
Ref | Expression |
---|---|
elfzonelfzo | ..^ ..^ ..^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzo2 9920 | . . 3 ..^ | |
2 | simpr 109 | . . . . . 6 ..^ | |
3 | eluzelz 9328 | . . . . . . . 8 | |
4 | 3 | 3ad2ant1 1002 | . . . . . . 7 |
5 | 4 | ad2antrr 479 | . . . . . 6 ..^ |
6 | 3 | adantr 274 | . . . . . . . . . . . . . . 15 |
7 | eluzel2 9324 | . . . . . . . . . . . . . . . 16 | |
8 | 7 | adantr 274 | . . . . . . . . . . . . . . 15 |
9 | simpr 109 | . . . . . . . . . . . . . . 15 | |
10 | elfzo 9919 | . . . . . . . . . . . . . . 15 ..^ | |
11 | 6, 8, 9, 10 | syl3anc 1216 | . . . . . . . . . . . . . 14 ..^ |
12 | eluzle 9331 | . . . . . . . . . . . . . . . 16 | |
13 | 12 | adantr 274 | . . . . . . . . . . . . . . 15 |
14 | 13 | biantrurd 303 | . . . . . . . . . . . . . 14 |
15 | 11, 14 | bitr4d 190 | . . . . . . . . . . . . 13 ..^ |
16 | 15 | notbid 656 | . . . . . . . . . . . 12 ..^ |
17 | 9 | zred 9166 | . . . . . . . . . . . . 13 |
18 | 6 | zred 9166 | . . . . . . . . . . . . 13 |
19 | 17, 18 | lenltd 7873 | . . . . . . . . . . . 12 |
20 | 16, 19 | bitr4d 190 | . . . . . . . . . . 11 ..^ |
21 | 20 | biimpd 143 | . . . . . . . . . 10 ..^ |
22 | 21 | ex 114 | . . . . . . . . 9 ..^ |
23 | 22 | com23 78 | . . . . . . . 8 ..^ |
24 | 23 | 3ad2ant1 1002 | . . . . . . 7 ..^ |
25 | 24 | imp31 254 | . . . . . 6 ..^ |
26 | eluz2 9325 | . . . . . 6 | |
27 | 2, 5, 25, 26 | syl3anbrc 1165 | . . . . 5 ..^ |
28 | simpll2 1021 | . . . . 5 ..^ | |
29 | simpll3 1022 | . . . . 5 ..^ | |
30 | elfzo2 9920 | . . . . 5 ..^ | |
31 | 27, 28, 29, 30 | syl3anbrc 1165 | . . . 4 ..^ ..^ |
32 | 31 | ex 114 | . . 3 ..^ ..^ |
33 | 1, 32 | sylanb 282 | . 2 ..^ ..^ ..^ |
34 | 33 | com12 30 | 1 ..^ ..^ ..^ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 962 wcel 1480 class class class wbr 3924 cfv 5118 (class class class)co 5767 clt 7793 cle 7794 cz 9047 cuz 9319 ..^cfzo 9912 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 df-fzo 9913 |
This theorem is referenced by: (None) |
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