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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 13330 | . 2 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏)}) | |
2 | df-icc 13331 | . 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 13128 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) | |
5 | 1, 2, 3, 4 | ixxssixx 13338 | 1 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∈ wcel 2107 ⊆ wss 3949 class class class wbr 5149 (class class class)co 7409 ℝ*cxr 11247 < clt 11248 ≤ cle 11249 [,)cico 13326 [,]cicc 13327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-ico 13330 df-icc 13331 |
This theorem is referenced by: iccpnfcnv 24460 itg2mulclem 25264 itg2mulc 25265 itg2monolem1 25268 itg2monolem2 25269 itg2monolem3 25270 itg2mono 25271 itg2i1fseq3 25275 itg2addlem 25276 itg2gt0 25278 itg2cnlem2 25280 psercnlem2 25936 eliccelico 31988 xrge0slmod 32463 xrge0iifcnv 32913 lmlimxrge0 32928 lmdvglim 32934 esumfsupre 33069 esumpfinvallem 33072 esumpfinval 33073 esumpfinvalf 33074 esumpcvgval 33076 esumpmono 33077 esummulc1 33079 sitmcl 33350 itg2addnc 36542 itg2gt0cn 36543 ftc1anclem6 36566 ftc1anclem8 36568 icoiccdif 44237 limciccioolb 44337 ltmod 44354 fourierdlem63 44885 fge0icoicc 45081 sge0tsms 45096 sge0iunmptlemre 45131 sge0isum 45143 sge0xaddlem1 45149 sge0xaddlem2 45150 sge0pnffsumgt 45158 sge0gtfsumgt 45159 sge0seq 45162 ovnsupge0 45273 ovnlecvr 45274 ovnsubaddlem1 45286 sge0hsphoire 45305 hoidmv1lelem3 45309 hoidmv1le 45310 hoidmvlelem1 45311 hoidmvlelem2 45312 hoidmvlelem3 45313 hoidmvlelem4 45314 hoidmvlelem5 45315 hoidmvle 45316 ovnhoilem1 45317 ovnlecvr2 45326 hspmbllem2 45343 sepfsepc 47560 |
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