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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 13393 | . 2 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏)}) | |
| 2 | df-icc 13394 | . 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 13191 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) | |
| 5 | 1, 2, 3, 4 | ixxssixx 13401 | 1 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 (class class class)co 7431 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 [,)cico 13389 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ico 13393 df-icc 13394 |
| This theorem is referenced by: iccpnfcnv 24975 itg2mulclem 25781 itg2mulc 25782 itg2monolem1 25785 itg2monolem2 25786 itg2monolem3 25787 itg2mono 25788 itg2i1fseq3 25792 itg2addlem 25793 itg2gt0 25795 itg2cnlem2 25797 psercnlem2 26468 eliccelico 32779 xrge0slmod 33376 xrge0iifcnv 33932 lmlimxrge0 33947 lmdvglim 33953 esumfsupre 34072 esumpfinvallem 34075 esumpfinval 34076 esumpfinvalf 34077 esumpcvgval 34079 esumpmono 34080 esummulc1 34082 sitmcl 34353 itg2addnc 37681 itg2gt0cn 37682 ftc1anclem6 37705 ftc1anclem8 37707 icoiccdif 45537 limciccioolb 45636 ltmod 45653 fourierdlem63 46184 fge0icoicc 46380 sge0tsms 46395 sge0iunmptlemre 46430 sge0isum 46442 sge0xaddlem1 46448 sge0xaddlem2 46449 sge0pnffsumgt 46457 sge0gtfsumgt 46458 sge0seq 46461 ovnsupge0 46572 ovnlecvr 46573 ovnsubaddlem1 46585 sge0hsphoire 46604 hoidmv1lelem3 46608 hoidmv1le 46609 hoidmvlelem1 46610 hoidmvlelem2 46611 hoidmvlelem3 46612 hoidmvlelem4 46613 hoidmvlelem5 46614 hoidmvle 46615 ovnhoilem1 46616 ovnlecvr2 46625 hspmbllem2 46642 sepfsepc 48825 |
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