Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ioogtlb | Structured version Visualization version GIF version |
Description: An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
ioogtlb | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioo2 12941 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
2 | simp2 1139 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 < 𝐶) | |
3 | 1, 2 | syl6bi 256 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) → 𝐴 < 𝐶)) |
4 | 3 | 3impia 1119 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2112 class class class wbr 5039 (class class class)co 7191 ℝcr 10693 ℝ*cxr 10831 < clt 10832 (,)cioo 12900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-pre-lttri 10768 ax-pre-lttrn 10769 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-ioo 12904 |
This theorem is referenced by: iocopn 42674 iooshift 42676 iooiinicc 42696 ioogtlbd 42704 iooiinioc 42710 lptre2pt 42799 limcresiooub 42801 limcresioolb 42802 sinaover2ne0 43027 dvbdfbdioolem1 43087 ioodvbdlimc1lem2 43091 fourierdlem27 43293 fourierdlem28 43294 fourierdlem31 43297 fourierdlem33 43299 fourierdlem40 43306 fourierdlem41 43307 fourierdlem46 43311 fourierdlem47 43312 fourierdlem48 43313 fourierdlem49 43314 fourierdlem57 43322 fourierdlem59 43324 fourierdlem62 43327 fourierdlem64 43329 fourierdlem65 43330 fourierdlem68 43333 fourierdlem73 43338 fourierdlem76 43341 fourierdlem78 43343 fourierdlem84 43349 fourierdlem90 43355 fourierdlem92 43357 fourierdlem97 43362 fourierdlem103 43368 fourierdlem104 43369 fourierdlem111 43376 sqwvfoura 43387 sqwvfourb 43388 fourierswlem 43389 fouriersw 43390 etransclem23 43416 qndenserrnbllem 43453 ioorrnopnlem 43463 ioorrnopnxrlem 43465 hoiqssbllem1 43778 hoiqssbllem2 43779 iunhoiioolem 43831 pimiooltgt 43863 smfaddlem1 43913 smfmullem1 43940 smfmullem2 43941 |
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