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Mirrors > Home > MPE Home > Th. List > fzossfz | Structured version Visualization version GIF version |
Description: A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
Ref | Expression |
---|---|
fzossfz | ⊢ (𝐴..^𝐵) ⊆ (𝐴...𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzofz 12903 | . 2 ⊢ (𝑥 ∈ (𝐴..^𝐵) → 𝑥 ∈ (𝐴...𝐵)) | |
2 | 1 | ssriv 3895 | 1 ⊢ (𝐴..^𝐵) ⊆ (𝐴...𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3861 (class class class)co 7019 ...cfz 12742 ..^cfzo 12883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-nn 11489 df-n0 11748 df-z 11832 df-uz 12094 df-fz 12743 df-fzo 12884 |
This theorem is referenced by: fzossz 12907 fzossnn0 12918 fzossnn 12936 elfzom1elp1fzo 12954 injresinjlem 13007 injresinj 13008 zmodfzp1 13113 uzindi 13200 wrdind 13920 wrd2ind 13921 scshwfzeqfzo 14024 telfsumo 14990 dfphi2 15940 cshwshashlem1 16258 psgnunilem5 18353 psgnunilem2 18354 efgredlemf 18594 efgredlemd 18597 efgredlemc 18598 uspgr2wlkeq 27110 wlkres 27134 redwlklem 27135 trlreslem 27163 pthdivtx 27192 eucrct2eupth 27706 cycpmfv2 30395 signstfvn 31448 signsvtn0 31449 breprexplemc 31512 fzossuz 41204 fourierdlem20 41968 fourierdlem25 41973 fourierdlem37 41985 fourierdlem64 42011 fourierdlem79 42026 fourierdlem89 42036 fourierdlem91 42038 fourierdlem101 42048 iccpartres 43074 iccpartipre 43077 iccpartleu 43084 bgoldbtbndlem2 43467 |
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