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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 12052 | . 2 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏)}) | |
2 | df-icc 12053 | . 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 11858 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) | |
5 | 1, 2, 3, 4 | ixxssixx 12060 | 1 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 (class class class)co 6549 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 [,)cico 12048 [,]cicc 12049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ico 12052 df-icc 12053 |
This theorem is referenced by: iccpnfcnv 22551 itg2mulclem 23319 itg2mulc 23320 itg2monolem1 23323 itg2monolem2 23324 itg2monolem3 23325 itg2mono 23326 itg2i1fseqle 23327 itg2i1fseq3 23330 itg2addlem 23331 itg2gt0 23333 itg2cnlem2 23335 psercnlem2 23982 eliccelico 28929 xrge0slmod 29175 xrge0iifcnv 29307 lmlimxrge0 29322 lmdvglim 29328 esumfsupre 29460 esumpfinvallem 29463 esumpfinval 29464 esumpfinvalf 29465 esumpcvgval 29467 esumpmono 29468 esummulc1 29470 sitmcl 29740 itg2addnc 32634 itg2gt0cn 32635 ftc1anclem6 32660 ftc1anclem8 32662 icoiccdif 38597 limciccioolb 38688 ltmod 38705 fourierdlem63 39062 fge0icoicc 39258 sge0tsms 39273 sge0iunmptlemre 39308 sge0isum 39320 sge0xaddlem1 39326 sge0xaddlem2 39327 sge0pnffsumgt 39335 sge0gtfsumgt 39336 sge0seq 39339 ovnsupge0 39447 ovnlecvr 39448 ovnsubaddlem1 39460 sge0hsphoire 39479 hoidmv1lelem3 39483 hoidmv1le 39484 hoidmvlelem1 39485 hoidmvlelem2 39486 hoidmvlelem3 39487 hoidmvlelem4 39488 hoidmvlelem5 39489 hoidmvle 39490 ovnhoilem1 39491 ovnlecvr2 39500 hspmbllem2 39517 |
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