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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 13271 | . 2 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏)}) | |
2 | df-icc 13272 | . 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 13069 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) | |
5 | 1, 2, 3, 4 | ixxssixx 13279 | 1 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∈ wcel 2107 ⊆ wss 3911 class class class wbr 5106 (class class class)co 7358 ℝ*cxr 11189 < clt 11190 ≤ cle 11191 [,)cico 13267 [,]cicc 13268 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-pre-lttri 11126 ax-pre-lttrn 11127 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-ico 13271 df-icc 13272 |
This theorem is referenced by: iccpnfcnv 24310 itg2mulclem 25114 itg2mulc 25115 itg2monolem1 25118 itg2monolem2 25119 itg2monolem3 25120 itg2mono 25121 itg2i1fseq3 25125 itg2addlem 25126 itg2gt0 25128 itg2cnlem2 25130 psercnlem2 25786 eliccelico 31683 xrge0slmod 32143 xrge0iifcnv 32517 lmlimxrge0 32532 lmdvglim 32538 esumfsupre 32673 esumpfinvallem 32676 esumpfinval 32677 esumpfinvalf 32678 esumpcvgval 32680 esumpmono 32681 esummulc1 32683 sitmcl 32954 itg2addnc 36135 itg2gt0cn 36136 ftc1anclem6 36159 ftc1anclem8 36161 icoiccdif 43769 limciccioolb 43869 ltmod 43886 fourierdlem63 44417 fge0icoicc 44613 sge0tsms 44628 sge0iunmptlemre 44663 sge0isum 44675 sge0xaddlem1 44681 sge0xaddlem2 44682 sge0pnffsumgt 44690 sge0gtfsumgt 44691 sge0seq 44694 ovnsupge0 44805 ovnlecvr 44806 ovnsubaddlem1 44818 sge0hsphoire 44837 hoidmv1lelem3 44841 hoidmv1le 44842 hoidmvlelem1 44843 hoidmvlelem2 44844 hoidmvlelem3 44845 hoidmvlelem4 44846 hoidmvlelem5 44847 hoidmvle 44848 ovnhoilem1 44849 ovnlecvr2 44858 hspmbllem2 44875 sepfsepc 46967 |
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