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Mirrors > Home > MPE Home > Th. List > ubioc1 | Structured version Visualization version GIF version |
Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 13439. (Contributed by FL, 29-May-2014.) |
Ref | Expression |
---|---|
ubioc1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ∈ (𝐴(,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1134 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ*) | |
2 | simp3 1135 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
3 | xrleid 13127 | . . 3 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵) | |
4 | 3 | 3ad2ant2 1131 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≤ 𝐵) |
5 | elioc1 13363 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵))) | |
6 | 5 | 3adant3 1129 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵))) |
7 | 1, 2, 4, 6 | mpbir3and 1339 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ∈ (𝐴(,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5138 (class class class)co 7401 ℝ*cxr 11244 < clt 11245 ≤ cle 11246 (,]cioc 13322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-ioc 13326 |
This theorem is referenced by: xrlimcnp 26816 pnfneige0 33420 lmxrge0 33421 bj-minftyccb 36596 dvasin 37062 dvacos 37063 limcresiooub 44843 xlimpnfvlem1 45037 fourierdlem79 45386 fourierdlem93 45400 fouriersw 45432 |
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