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Mirrors > Home > MPE Home > Th. List > iooss2 | Structured version Visualization version GIF version |
Description: Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
iooss2 | ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ≤ 𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ioo 12592 | . 2 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | xrltletr 12400 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝑤 < 𝐶)) | |
3 | 1, 1, 2 | ixxss2 12607 | 1 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ≤ 𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2081 ⊆ wss 3859 class class class wbr 4962 (class class class)co 7016 ℝ*cxr 10520 < clt 10521 ≤ cle 10522 (,)cioo 12588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-pre-lttri 10457 ax-pre-lttrn 10458 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-po 5362 df-so 5363 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-ioo 12592 |
This theorem is referenced by: tgqioo 23091 ioorcl2 23856 itgsplitioo 24121 ditgcl 24139 ditgswap 24140 ditgsplitlem 24141 dvferm2lem 24266 dvferm 24268 dvlip 24273 dvgt0lem1 24282 dvivthlem1 24288 lhop1lem 24293 lhop1 24294 dvcvx 24300 dvfsumle 24301 dvfsumge 24302 dvfsumabs 24303 ftc1lem1 24315 ftc1lem2 24316 ftc1a 24317 ftc1lem4 24319 ftc2 24324 ftc2ditglem 24325 itgsubstlem 24328 ftc1anc 34506 ftc2nc 34507 limcresioolb 41466 fourierdlem46 41979 fourierdlem48 41981 fourierdlem49 41982 fourierdlem75 42008 fourierdlem103 42036 fourierdlem113 42046 fouriersw 42058 |
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