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Mirrors > Home > MPE Home > Th. List > icossicc | Structured version Visualization version GIF version |
Description: A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.) |
Ref | Expression |
---|---|
icossicc | ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ico 13326 | . 2 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏)}) | |
2 | df-icc 13327 | . 2 ⊢ [,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 ≤ 𝑏)}) | |
3 | idd 24 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴 ≤ 𝑤 → 𝐴 ≤ 𝑤)) | |
4 | xrltle 13124 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) | |
5 | 1, 2, 3, 4 | ixxssixx 13334 | 1 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∈ wcel 2106 ⊆ wss 3947 class class class wbr 5147 (class class class)co 7405 ℝ*cxr 11243 < clt 11244 ≤ cle 11245 [,)cico 13322 [,]cicc 13323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-ico 13326 df-icc 13327 |
This theorem is referenced by: iccpnfcnv 24451 itg2mulclem 25255 itg2mulc 25256 itg2monolem1 25259 itg2monolem2 25260 itg2monolem3 25261 itg2mono 25262 itg2i1fseq3 25266 itg2addlem 25267 itg2gt0 25269 itg2cnlem2 25271 psercnlem2 25927 eliccelico 31975 xrge0slmod 32451 xrge0iifcnv 32901 lmlimxrge0 32916 lmdvglim 32922 esumfsupre 33057 esumpfinvallem 33060 esumpfinval 33061 esumpfinvalf 33062 esumpcvgval 33064 esumpmono 33065 esummulc1 33067 sitmcl 33338 itg2addnc 36530 itg2gt0cn 36531 ftc1anclem6 36554 ftc1anclem8 36556 icoiccdif 44223 limciccioolb 44323 ltmod 44340 fourierdlem63 44871 fge0icoicc 45067 sge0tsms 45082 sge0iunmptlemre 45117 sge0isum 45129 sge0xaddlem1 45135 sge0xaddlem2 45136 sge0pnffsumgt 45144 sge0gtfsumgt 45145 sge0seq 45148 ovnsupge0 45259 ovnlecvr 45260 ovnsubaddlem1 45272 sge0hsphoire 45291 hoidmv1lelem3 45295 hoidmv1le 45296 hoidmvlelem1 45297 hoidmvlelem2 45298 hoidmvlelem3 45299 hoidmvlelem4 45300 hoidmvlelem5 45301 hoidmvle 45302 ovnhoilem1 45303 ovnlecvr2 45312 hspmbllem2 45329 sepfsepc 47513 |
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