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Mirrors > Home > MPE Home > Th. List > elico1 | Structured version Visualization version GIF version |
Description: Membership in a closed-below, open-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
elico1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ico 13390 | . 2 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | 1 | elixx1 13393 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 [,)cico 13386 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xr 11297 df-ico 13390 |
This theorem is referenced by: elicod 13434 icogelb 13435 lbico1 13438 elico2 13448 icodisj 13513 ico01fl0 13856 addmodid 13957 leordtvallem2 23235 pnfnei 23244 mnfnei 23245 blval2 24591 metuel2 24594 iscfil2 25314 eliccelico 32786 elicoelioo 32787 xrdifh 32789 fsumrp0cl 33009 ply1degltel 33595 ply1degleel 33596 ply1degltdimlem 33650 xrge0iifcnv 33894 esumpcvgval 34059 dnizeq0 36458 relowlssretop 37346 tan2h 37599 iocinico 43201 rfcnpre3 44971 icoltub 45461 icoiccdif 45477 iccelpart 47358 icceuelpart 47361 bgoldbtbndlem1 47730 bgoldbtbndlem2 47731 bgoldbtbndlem3 47732 bgoldbtbnd 47734 |
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