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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 9927 | . . 3 ..^ | |
2 | simpr 109 | . . . . . 6 ..^ | |
3 | eluzelz 9335 | . . . . . . . 8 | |
4 | 3 | 3ad2ant1 1002 | . . . . . . 7 |
5 | 4 | ad2antrr 479 | . . . . . 6 ..^ |
6 | 3 | adantr 274 | . . . . . . . . . . . . . . 15 |
7 | eluzel2 9331 | . . . . . . . . . . . . . . . 16 | |
8 | 7 | adantr 274 | . . . . . . . . . . . . . . 15 |
9 | simpr 109 | . . . . . . . . . . . . . . 15 | |
10 | elfzo 9926 | . . . . . . . . . . . . . . 15 ..^ | |
11 | 6, 8, 9, 10 | syl3anc 1216 | . . . . . . . . . . . . . 14 ..^ |
12 | eluzle 9338 | . . . . . . . . . . . . . . . 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 9173 | . . . . . . . . . . . . 13 |
18 | 6 | zred 9173 | . . . . . . . . . . . . 13 |
19 | 17, 18 | lenltd 7880 | . . . . . . . . . . . 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 9332 | . . . . . 6 | |
27 | 2, 5, 25, 26 | syl3anbrc 1165 | . . . . 5 ..^ |
28 | simpll2 1021 | . . . . 5 ..^ | |
29 | simpll3 1022 | . . . . 5 ..^ | |
30 | elfzo2 9927 | . . . . 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 3929 cfv 5123 (class class class)co 5774 clt 7800 cle 7801 cz 9054 cuz 9326 ..^cfzo 9919 |
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-csb 3004 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-iun 3815 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-1st 6038 df-2nd 6039 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 df-fzo 9920 |
This theorem is referenced by: (None) |
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