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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iocleub | Structured version Visualization version GIF version | ||
| Description: An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| iocleub | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≤ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elioc1 13425 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
| 2 | simp3 1139 | . . 3 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ≤ 𝐵) | |
| 3 | 1, 2 | biimtrdi 253 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) → 𝐶 ≤ 𝐵)) |
| 4 | 3 | 3impia 1118 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≤ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5141 (class class class)co 7429 ℝ*cxr 11290 < clt 11291 ≤ cle 11292 (,]cioc 13384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6512 df-fun 6561 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-xr 11295 df-ioc 13388 |
| This theorem is referenced by: iocopn 45506 iccdificc 45525 iocleubd 45545 limcresiooub 45630 fourierdlem19 46114 fourierdlem35 46130 fourierdlem41 46136 fourierdlem46 46140 fourierdlem48 46142 fourierdlem49 46143 fourierswlem 46218 fouriersw 46219 pimdecfgtioc 46703 pimincfltioc 46704 |
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