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Mathbox for Glauco Siliprandi |
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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 13419 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
2 | simp3 1136 | . . 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 1115 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2104 class class class wbr 5149 (class class class)co 7425 ℝ*cxr 11285 < clt 11286 ≤ cle 11287 (,]cioc 13378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-resscn 11203 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6510 df-fun 6560 df-fv 6566 df-ov 7428 df-oprab 7429 df-mpo 7430 df-xr 11290 df-ioc 13382 |
This theorem is referenced by: iocopn 45423 iccdificc 45442 iocleubd 45462 limcresiooub 45548 fourierdlem19 46032 fourierdlem35 46048 fourierdlem41 46054 fourierdlem46 46058 fourierdlem48 46060 fourierdlem49 46061 fourierswlem 46136 fouriersw 46137 pimdecfgtioc 46621 pimincfltioc 46622 |
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