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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 10075 | . . 3 ..^ | |
2 | simpr 109 | . . . . . 6 ..^ | |
3 | eluzelz 9466 | . . . . . . . 8 | |
4 | 3 | 3ad2ant1 1007 | . . . . . . 7 |
5 | 4 | ad2antrr 480 | . . . . . 6 ..^ |
6 | 3 | adantr 274 | . . . . . . . . . . . . . . 15 |
7 | eluzel2 9462 | . . . . . . . . . . . . . . . 16 | |
8 | 7 | adantr 274 | . . . . . . . . . . . . . . 15 |
9 | simpr 109 | . . . . . . . . . . . . . . 15 | |
10 | elfzo 10074 | . . . . . . . . . . . . . . 15 ..^ | |
11 | 6, 8, 9, 10 | syl3anc 1227 | . . . . . . . . . . . . . 14 ..^ |
12 | eluzle 9469 | . . . . . . . . . . . . . . . 16 | |
13 | 12 | adantr 274 | . . . . . . . . . . . . . . 15 |
14 | 13 | biantrurd 303 | . . . . . . . . . . . . . 14 |
15 | 11, 14 | bitr4d 190 | . . . . . . . . . . . . 13 ..^ |
16 | 15 | notbid 657 | . . . . . . . . . . . 12 ..^ |
17 | 9 | zred 9304 | . . . . . . . . . . . . 13 |
18 | 6 | zred 9304 | . . . . . . . . . . . . 13 |
19 | 17, 18 | lenltd 8007 | . . . . . . . . . . . 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 1007 | . . . . . . 7 ..^ |
25 | 24 | imp31 254 | . . . . . 6 ..^ |
26 | eluz2 9463 | . . . . . 6 | |
27 | 2, 5, 25, 26 | syl3anbrc 1170 | . . . . 5 ..^ |
28 | simpll2 1026 | . . . . 5 ..^ | |
29 | simpll3 1027 | . . . . 5 ..^ | |
30 | elfzo2 10075 | . . . . 5 ..^ | |
31 | 27, 28, 29, 30 | syl3anbrc 1170 | . . . 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 967 wcel 2135 class class class wbr 3976 cfv 5182 (class class class)co 5836 clt 7924 cle 7925 cz 9182 cuz 9457 ..^cfzo 10067 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 df-fzo 10068 |
This theorem is referenced by: (None) |
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