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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 12593 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
2 | simp2 1118 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 < 𝐶) | |
3 | 1, 2 | syl6bi 245 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) → 𝐴 < 𝐶)) |
4 | 3 | 3impia 1098 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1069 ∈ wcel 2051 class class class wbr 4925 (class class class)co 6974 ℝcr 10332 ℝ*cxr 10471 < clt 10472 (,)cioo 12552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-pre-lttri 10407 ax-pre-lttrn 10408 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-po 5322 df-so 5323 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-1st 7499 df-2nd 7500 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-ioo 12556 |
This theorem is referenced by: iocopn 41261 iooshift 41263 iooiinicc 41283 ioogtlbd 41291 iooiinioc 41297 lptre2pt 41386 limcresiooub 41388 limcresioolb 41389 sinaover2ne0 41613 dvbdfbdioolem1 41677 ioodvbdlimc1lem2 41681 fourierdlem27 41884 fourierdlem28 41885 fourierdlem31 41888 fourierdlem33 41890 fourierdlem40 41897 fourierdlem41 41898 fourierdlem46 41902 fourierdlem47 41903 fourierdlem48 41904 fourierdlem49 41905 fourierdlem57 41913 fourierdlem59 41915 fourierdlem60 41916 fourierdlem61 41917 fourierdlem62 41918 fourierdlem64 41920 fourierdlem65 41921 fourierdlem68 41924 fourierdlem73 41929 fourierdlem76 41932 fourierdlem78 41934 fourierdlem84 41940 fourierdlem90 41946 fourierdlem92 41948 fourierdlem97 41953 fourierdlem103 41959 fourierdlem104 41960 fourierdlem111 41967 sqwvfoura 41978 sqwvfourb 41979 fourierswlem 41980 fouriersw 41981 etransclem23 42007 qndenserrnbllem 42044 ioorrnopnlem 42054 ioorrnopnxrlem 42056 hoiqssbllem1 42369 hoiqssbllem2 42370 iunhoiioolem 42422 pimiooltgt 42454 smfaddlem1 42504 smfmullem1 42531 smfmullem2 42532 |
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