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Mirrors > Home > ILE Home > Th. List > iooss2 | 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 9876 | . 2 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | xrltletr 9791 | . 2 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝑤 < 𝐶)) | |
3 | 1, 1, 2 | ixxss2 9889 | 1 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ≤ 𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 ⊆ wss 3129 class class class wbr 4000 (class class class)co 5869 ℝ*cxr 7978 < clt 7979 ≤ cle 7980 (,)cioo 9872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-po 4293 df-iso 4294 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-ioo 9876 |
This theorem is referenced by: tgqioo 13707 cos0pilt1 13933 |
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