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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 13384 | . 2 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏)}) | |
2 | df-icc 13385 | . 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 13182 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) | |
5 | 1, 2, 3, 4 | ixxssixx 13392 | 1 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∈ wcel 2099 ⊆ wss 3947 class class class wbr 5153 (class class class)co 7424 ℝ*cxr 11297 < clt 11298 ≤ cle 11299 [,)cico 13380 [,]cicc 13381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-pre-lttri 11232 ax-pre-lttrn 11233 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-ico 13384 df-icc 13385 |
This theorem is referenced by: iccpnfcnv 24960 itg2mulclem 25767 itg2mulc 25768 itg2monolem1 25771 itg2monolem2 25772 itg2monolem3 25773 itg2mono 25774 itg2i1fseq3 25778 itg2addlem 25779 itg2gt0 25781 itg2cnlem2 25783 psercnlem2 26454 eliccelico 32679 xrge0slmod 33223 xrge0iifcnv 33748 lmlimxrge0 33763 lmdvglim 33769 esumfsupre 33904 esumpfinvallem 33907 esumpfinval 33908 esumpfinvalf 33909 esumpcvgval 33911 esumpmono 33912 esummulc1 33914 sitmcl 34185 itg2addnc 37375 itg2gt0cn 37376 ftc1anclem6 37399 ftc1anclem8 37401 icoiccdif 45142 limciccioolb 45242 ltmod 45259 fourierdlem63 45790 fge0icoicc 45986 sge0tsms 46001 sge0iunmptlemre 46036 sge0isum 46048 sge0xaddlem1 46054 sge0xaddlem2 46055 sge0pnffsumgt 46063 sge0gtfsumgt 46064 sge0seq 46067 ovnsupge0 46178 ovnlecvr 46179 ovnsubaddlem1 46191 sge0hsphoire 46210 hoidmv1lelem3 46214 hoidmv1le 46215 hoidmvlelem1 46216 hoidmvlelem2 46217 hoidmvlelem3 46218 hoidmvlelem4 46219 hoidmvlelem5 46220 hoidmvle 46221 ovnhoilem1 46222 ovnlecvr2 46231 hspmbllem2 46248 sepfsepc 48261 |
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