![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > elfzoelz | GIF version |
Description: Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
Ref | Expression |
---|---|
elfzoelz | β’ (π΄ β (π΅..^πΆ) β π΄ β β€) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzoel1 10145 | . . . 4 β’ (π΄ β (π΅..^πΆ) β π΅ β β€) | |
2 | elfzoel2 10146 | . . . 4 β’ (π΄ β (π΅..^πΆ) β πΆ β β€) | |
3 | fzof 10144 | . . . . 5 β’ ..^:(β€ Γ β€)βΆπ« β€ | |
4 | 3 | fovcl 5980 | . . . 4 β’ ((π΅ β β€ β§ πΆ β β€) β (π΅..^πΆ) β π« β€) |
5 | 1, 2, 4 | syl2anc 411 | . . 3 β’ (π΄ β (π΅..^πΆ) β (π΅..^πΆ) β π« β€) |
6 | 5 | elpwid 3587 | . 2 β’ (π΄ β (π΅..^πΆ) β (π΅..^πΆ) β β€) |
7 | id 19 | . 2 β’ (π΄ β (π΅..^πΆ) β π΄ β (π΅..^πΆ)) | |
8 | 6, 7 | sseldd 3157 | 1 β’ (π΄ β (π΅..^πΆ) β π΄ β β€) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β wcel 2148 π« cpw 3576 (class class class)co 5875 β€cz 9253 ..^cfzo 10142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-addcom 7911 ax-addass 7913 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-0id 7919 ax-rnegex 7920 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-ltadd 7927 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-inn 8920 df-n0 9177 df-z 9254 df-fz 10009 df-fzo 10143 |
This theorem is referenced by: elfzo2 10150 elfzole1 10155 elfzolt2 10156 elfzolt3 10157 elfzolt2b 10158 elfzouz2 10161 fzonnsub 10169 fzospliti 10176 fzodisj 10178 fzonmapblen 10187 fzoaddel 10192 fzosubel 10194 modaddmodup 10387 modaddmodlo 10388 modfzo0difsn 10395 modsumfzodifsn 10396 addmodlteq 10398 iseqf1olemqk 10494 seq3f1olemp 10502 fzomaxdiflem 11121 fzomaxdif 11122 fzo0dvdseq 11863 fzocongeq 11864 addmodlteqALT 11865 crth 12224 phimullem 12225 eulerthlem1 12227 eulerthlemfi 12228 eulerthlemrprm 12229 hashgcdlem 12238 hashgcdeq 12239 phisum 12240 reumodprminv 12253 modprm0 12254 nnnn0modprm0 12255 modprmn0modprm0 12256 nninfdclemlt 12452 trilpolemeq1 14791 |
Copyright terms: Public domain | W3C validator |