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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 13334 | . 2 ⊢ [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏)}) | |
2 | df-icc 13335 | . 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 13132 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵)) | |
5 | 1, 2, 3, 4 | ixxssixx 13342 | 1 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∈ wcel 2104 ⊆ wss 3947 class class class wbr 5147 (class class class)co 7411 ℝ*cxr 11251 < clt 11252 ≤ cle 11253 [,)cico 13330 [,]cicc 13331 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-ico 13334 df-icc 13335 |
This theorem is referenced by: iccpnfcnv 24689 itg2mulclem 25496 itg2mulc 25497 itg2monolem1 25500 itg2monolem2 25501 itg2monolem3 25502 itg2mono 25503 itg2i1fseq3 25507 itg2addlem 25508 itg2gt0 25510 itg2cnlem2 25512 psercnlem2 26172 eliccelico 32255 xrge0slmod 32733 xrge0iifcnv 33211 lmlimxrge0 33226 lmdvglim 33232 esumfsupre 33367 esumpfinvallem 33370 esumpfinval 33371 esumpfinvalf 33372 esumpcvgval 33374 esumpmono 33375 esummulc1 33377 sitmcl 33648 itg2addnc 36845 itg2gt0cn 36846 ftc1anclem6 36869 ftc1anclem8 36871 icoiccdif 44535 limciccioolb 44635 ltmod 44652 fourierdlem63 45183 fge0icoicc 45379 sge0tsms 45394 sge0iunmptlemre 45429 sge0isum 45441 sge0xaddlem1 45447 sge0xaddlem2 45448 sge0pnffsumgt 45456 sge0gtfsumgt 45457 sge0seq 45460 ovnsupge0 45571 ovnlecvr 45572 ovnsubaddlem1 45584 sge0hsphoire 45603 hoidmv1lelem3 45607 hoidmv1le 45608 hoidmvlelem1 45609 hoidmvlelem2 45610 hoidmvlelem3 45611 hoidmvlelem4 45612 hoidmvlelem5 45613 hoidmvle 45614 ovnhoilem1 45615 ovnlecvr2 45624 hspmbllem2 45641 sepfsepc 47647 |
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