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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 12773 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
2 | simp2 1133 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 < 𝐶) | |
3 | 1, 2 | syl6bi 255 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) → 𝐴 < 𝐶)) |
4 | 3 | 3impia 1113 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 ℝ*cxr 10668 < clt 10669 (,)cioo 12732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-ioo 12736 |
This theorem is referenced by: iocopn 41789 iooshift 41791 iooiinicc 41811 ioogtlbd 41819 iooiinioc 41825 lptre2pt 41914 limcresiooub 41916 limcresioolb 41917 sinaover2ne0 42142 dvbdfbdioolem1 42206 ioodvbdlimc1lem2 42210 fourierdlem27 42413 fourierdlem28 42414 fourierdlem31 42417 fourierdlem33 42419 fourierdlem40 42426 fourierdlem41 42427 fourierdlem46 42431 fourierdlem47 42432 fourierdlem48 42433 fourierdlem49 42434 fourierdlem57 42442 fourierdlem59 42444 fourierdlem62 42447 fourierdlem64 42449 fourierdlem65 42450 fourierdlem68 42453 fourierdlem73 42458 fourierdlem76 42461 fourierdlem78 42463 fourierdlem84 42469 fourierdlem90 42475 fourierdlem92 42477 fourierdlem97 42482 fourierdlem103 42488 fourierdlem104 42489 fourierdlem111 42496 sqwvfoura 42507 sqwvfourb 42508 fourierswlem 42509 fouriersw 42510 etransclem23 42536 qndenserrnbllem 42573 ioorrnopnlem 42583 ioorrnopnxrlem 42585 hoiqssbllem1 42898 hoiqssbllem2 42899 iunhoiioolem 42951 pimiooltgt 42983 smfaddlem1 43033 smfmullem1 43060 smfmullem2 43061 |
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