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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 12730 | . 2 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | xrltletr 12538 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝑤 < 𝐶)) | |
3 | 1, 1, 2 | ixxss2 12745 | 1 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ≤ 𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 class class class wbr 5057 (class class class)co 7145 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 (,)cioo 12726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-ioo 12730 |
This theorem is referenced by: tgqioo 23335 ioorcl2 24100 itgsplitioo 24365 ditgcl 24383 ditgswap 24384 ditgsplitlem 24385 dvferm2lem 24510 dvferm 24512 dvlip 24517 dvgt0lem1 24526 dvivthlem1 24532 lhop1lem 24537 lhop1 24538 dvcvx 24544 dvfsumle 24545 dvfsumge 24546 dvfsumabs 24547 ftc1lem1 24559 ftc1lem2 24560 ftc1a 24561 ftc1lem4 24563 ftc2 24568 ftc2ditglem 24569 itgsubstlem 24572 ftc1anc 34856 ftc2nc 34857 limcresioolb 41800 fourierdlem46 42314 fourierdlem48 42316 fourierdlem49 42317 fourierdlem75 42343 fourierdlem103 42371 fourierdlem113 42381 fouriersw 42393 |
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