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Mirrors > Home > ILE Home > Th. List > fzouzsplit | Unicode version |
Description: Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.) |
Ref | Expression |
---|---|
fzouzsplit | ..^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 9448 | . . . . . . . 8 | |
2 | eluzelz 9448 | . . . . . . . 8 | |
3 | zlelttric 9212 | . . . . . . . 8 | |
4 | 1, 2, 3 | syl2an 287 | . . . . . . 7 |
5 | 4 | orcomd 719 | . . . . . 6 |
6 | id 19 | . . . . . . . 8 | |
7 | elfzo2 10049 | . . . . . . . . . 10 ..^ | |
8 | df-3an 965 | . . . . . . . . . 10 | |
9 | 7, 8 | bitri 183 | . . . . . . . . 9 ..^ |
10 | 9 | baib 905 | . . . . . . . 8 ..^ |
11 | 6, 1, 10 | syl2anr 288 | . . . . . . 7 ..^ |
12 | eluz 9452 | . . . . . . . 8 | |
13 | 1, 2, 12 | syl2an 287 | . . . . . . 7 |
14 | 11, 13 | orbi12d 783 | . . . . . 6 ..^ |
15 | 5, 14 | mpbird 166 | . . . . 5 ..^ |
16 | 15 | ex 114 | . . . 4 ..^ |
17 | elun 3248 | . . . 4 ..^ ..^ | |
18 | 16, 17 | syl6ibr 161 | . . 3 ..^ |
19 | 18 | ssrdv 3134 | . 2 ..^ |
20 | elfzouz 10050 | . . . . 5 ..^ | |
21 | 20 | ssriv 3132 | . . . 4 ..^ |
22 | 21 | a1i 9 | . . 3 ..^ |
23 | uzss 9459 | . . 3 | |
24 | 22, 23 | unssd 3283 | . 2 ..^ |
25 | 19, 24 | eqssd 3145 | 1 ..^ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 w3a 963 wceq 1335 wcel 2128 cun 3100 wss 3102 class class class wbr 3965 cfv 5170 (class class class)co 5824 clt 7912 cle 7913 cz 9167 cuz 9439 ..^cfzo 10041 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 df-n0 9091 df-z 9168 df-uz 9440 df-fz 9913 df-fzo 10042 |
This theorem is referenced by: zsupcllemstep 11831 |
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