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Mirrors > Home > MPE Home > Th. List > icoval | Structured version Visualization version GIF version |
Description: Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
icoval | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ico 12770 | . 2 ⊢ [,) = (𝑦 ∈ ℝ*, 𝑧 ∈ ℝ* ↦ {𝑥 ∈ ℝ* ∣ (𝑦 ≤ 𝑥 ∧ 𝑥 < 𝑧)}) | |
2 | 1 | ixxval 12772 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 {crab 3072 class class class wbr 5025 (class class class)co 7143 ℝ*cxr 10697 < clt 10698 ≤ cle 10699 [,)cico 12766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-iota 6287 df-fun 6330 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-xr 10702 df-ico 12770 |
This theorem is referenced by: ico0 12810 orvcgteel 31938 elicores 42521 volicorescl 43543 |
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