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Mirrors > Home > MPE Home > Th. List > icossre | Structured version Visualization version GIF version |
Description: A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.) |
Ref | Expression |
---|---|
icossre | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elico2 12430 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴[,)𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))) | |
2 | 1 | biimp3a 1581 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,)𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)) |
3 | 2 | simp1d 1137 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,)𝐵)) → 𝑥 ∈ ℝ) |
4 | 3 | 3expia 1115 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴[,)𝐵) → 𝑥 ∈ ℝ)) |
5 | 4 | ssrdv 3750 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 ∈ wcel 2139 ⊆ wss 3715 class class class wbr 4804 (class class class)co 6813 ℝcr 10127 ℝ*cxr 10265 < clt 10266 ≤ cle 10267 [,)cico 12370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-pre-lttri 10202 ax-pre-lttrn 10203 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-ico 12374 |
This theorem is referenced by: icoshftf1o 12488 ico01fl0 12814 rexico 14292 rlim3 14428 fprodge1 14925 ovolicopnf 23492 dvfsumrlim2 23994 tanord1 24482 chebbnd1 25360 chebbnd2 25365 dchrisumlem3 25379 pntpbnd1 25474 pntibndlem2 25479 sxbrsigalem0 30642 dya2iocress 30645 dya2iocucvr 30655 sitmcl 30722 tan2h 33714 icoopn 40254 limciccioolb 40356 ltmod 40373 limcresioolb 40378 limsupresre 40431 limsupresico 40435 liminfresico 40506 fourierdlem32 40859 fourierdlem46 40872 fourierdlem48 40874 fourierdlem93 40919 fouriersw 40951 fouriercn 40952 hoissre 41264 hoissrrn2 41298 hoidmv1lelem2 41312 ovnlecvr2 41330 hspdifhsp 41336 hoiqssbllem2 41343 hspmbllem2 41347 iinhoiicclem 41393 |
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