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Theorem icoval 12155
 Description: Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
icoval ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴𝑥𝑥 < 𝐵)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem icoval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ico 12123 . 2 [,) = (𝑦 ∈ ℝ*, 𝑧 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑦𝑥𝑥 < 𝑧)})
21ixxval 12125 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴𝑥𝑥 < 𝐵)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  {crab 2911   class class class wbr 4613  (class class class)co 6604  ℝ*cxr 10017   < clt 10018   ≤ cle 10019  [,)cico 12119 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-xr 10022  df-ico 12123 This theorem is referenced by:  ico0  12163  orvcgteel  30310  elicores  39171  volicorescl  40074
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