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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 13391 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
2 | simp2 1135 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 < 𝐶) | |
3 | 1, 2 | syl6bi 253 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) → 𝐴 < 𝐶)) |
4 | 3 | 3impia 1115 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2099 class class class wbr 5142 (class class class)co 7414 ℝcr 11131 ℝ*cxr 11271 < clt 11272 (,)cioo 13350 |
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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-pre-lttri 11206 ax-pre-lttrn 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-ioo 13354 |
This theorem is referenced by: iocopn 44899 iooshift 44901 iooiinicc 44921 ioogtlbd 44929 iooiinioc 44935 lptre2pt 45022 limcresiooub 45024 limcresioolb 45025 sinaover2ne0 45250 dvbdfbdioolem1 45310 ioodvbdlimc1lem2 45314 fourierdlem27 45516 fourierdlem28 45517 fourierdlem31 45520 fourierdlem33 45522 fourierdlem40 45529 fourierdlem41 45530 fourierdlem46 45534 fourierdlem47 45535 fourierdlem48 45536 fourierdlem49 45537 fourierdlem57 45545 fourierdlem59 45547 fourierdlem62 45550 fourierdlem64 45552 fourierdlem65 45553 fourierdlem68 45556 fourierdlem73 45561 fourierdlem76 45564 fourierdlem78 45566 fourierdlem84 45572 fourierdlem90 45578 fourierdlem92 45580 fourierdlem97 45585 fourierdlem103 45591 fourierdlem104 45592 fourierdlem111 45599 sqwvfoura 45610 sqwvfourb 45611 fourierswlem 45612 fouriersw 45613 etransclem23 45639 qndenserrnbllem 45676 ioorrnopnlem 45686 ioorrnopnxrlem 45688 hoiqssbllem1 46004 hoiqssbllem2 46005 iunhoiioolem 46057 pimiooltgt 46092 smfaddlem1 46145 smfmullem1 46173 smfmullem2 46174 |
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