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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 12794 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴[,)𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))) | |
2 | 1 | biimp3a 1465 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,)𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)) |
3 | 2 | simp1d 1138 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,)𝐵)) → 𝑥 ∈ ℝ) |
4 | 3 | 3expia 1117 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝑥 ∈ (𝐴[,)𝐵) → 𝑥 ∈ ℝ)) |
5 | 4 | ssrdv 3972 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 [,)cico 12734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-ico 12738 |
This theorem is referenced by: icoshftf1o 12854 ico01fl0 13183 rexico 14707 rlim3 14849 fprodge1 15343 ovolicopnf 24119 dvfsumrlim2 24623 tanord1 25115 chebbnd1 26042 chebbnd2 26047 dchrisumlem3 26061 pntpbnd1 26156 pntibndlem2 26161 sxbrsigalem0 31524 dya2iocress 31527 dya2iocucvr 31537 sitmcl 31604 tan2h 34878 icoopn 41794 limciccioolb 41895 ltmod 41912 limcresioolb 41917 limsupresre 41970 limsupresico 41974 liminfresico 42045 fourierdlem32 42418 fourierdlem46 42431 fourierdlem48 42433 fourierdlem93 42478 fouriersw 42510 fouriercn 42511 hoissre 42820 hoissrrn2 42854 hoidmv1lelem2 42868 ovnlecvr2 42886 hspdifhsp 42892 hoiqssbllem2 42899 hspmbllem2 42903 iinhoiicclem 42949 |
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