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| 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 | ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfzoel1 10249 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) | |
| 2 | elfzoel2 10250 | . . . 4 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) | |
| 3 | fzof 10248 | . . . . 5 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | |
| 4 | 3 | fovcl 6041 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵..^𝐶) ∈ 𝒫 ℤ) |
| 5 | 1, 2, 4 | syl2anc 411 | . . 3 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ∈ 𝒫 ℤ) |
| 6 | 5 | elpwid 3626 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → (𝐵..^𝐶) ⊆ ℤ) |
| 7 | id 19 | . 2 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ (𝐵..^𝐶)) | |
| 8 | 6, 7 | sseldd 3193 | 1 ⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∈ wcel 2175 𝒫 cpw 3615 (class class class)co 5934 ℤcz 9354 ..^cfzo 10246 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-addcom 8007 ax-addass 8009 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-0id 8015 ax-rnegex 8016 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-ltadd 8023 |
| This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4338 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-fv 5276 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-1st 6216 df-2nd 6217 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-inn 9019 df-n0 9278 df-z 9355 df-fz 10113 df-fzo 10247 |
| This theorem is referenced by: elfzo2 10254 elfzole1 10260 elfzolt2 10261 elfzolt3 10262 elfzolt2b 10263 elfzouz2 10266 fzonnsub 10274 fzospliti 10281 fzodisj 10283 fzonmapblen 10292 fzoaddel 10297 elincfzoext 10303 fzosubel 10304 modaddmodup 10513 modaddmodlo 10514 modfzo0difsn 10521 modsumfzodifsn 10522 addmodlteq 10524 iseqf1olemqk 10633 seq3f1olemp 10641 seqfeq4g 10657 ccatcl 11024 ccatlen 11026 ccatval2 11029 ccatval3 11030 ccatvalfn 11032 ccatlid 11037 ccatass 11039 ccatrn 11040 fzomaxdiflem 11342 fzomaxdif 11343 fzo0dvdseq 12087 fzocongeq 12088 addmodlteqALT 12089 crth 12465 phimullem 12466 eulerthlem1 12468 eulerthlemfi 12469 eulerthlemrprm 12470 hashgcdlem 12479 hashgcdeq 12481 phisum 12482 reumodprminv 12495 modprm0 12496 nnnn0modprm0 12497 modprmn0modprm0 12498 4sqlemafi 12637 nninfdclemlt 12741 gsumfzfsumlemm 14267 znf1o 14331 trilpolemeq1 15843 |
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