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Mirrors > Home > MPE Home > Th. List > iooss1 | 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, 20-Feb-2015.) |
Ref | Expression |
---|---|
iooss1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ioo 12179 | . 2 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | xrlelttr 11987 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝑤) → 𝐴 < 𝑤)) | |
3 | 1, 1, 2 | ixxss1 12193 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ⊆ wss 3574 class class class wbr 4653 (class class class)co 6650 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 (,)cioo 12175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioo 12179 |
This theorem is referenced by: ioodisj 12302 tgqioo 22603 ioorcl2 23340 itg2gt0 23527 itgsplitioo 23604 ditgcl 23622 ditgswap 23623 ditgsplitlem 23624 dvferm1lem 23747 dvferm 23751 dvlip 23756 dvgt0lem1 23765 dvivthlem1 23771 lhop1lem 23776 lhop2 23778 dvcvx 23783 dvfsumle 23784 dvfsumge 23785 dvfsumabs 23786 ftc1lem1 23798 ftc1a 23800 ftc1lem4 23802 ftc2ditglem 23808 tanregt0 24285 basellem4 24810 pntlemp 25299 radcnvrat 38513 limcresiooub 39874 fourierdlem46 40369 fourierdlem48 40371 fourierdlem49 40372 fourierdlem74 40397 fourierdlem104 40427 fourierdlem113 40436 fouriersw 40448 |
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