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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iooltub | Structured version Visualization version GIF version |
Description: An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
iooltub | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 < 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioo2 13297 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
2 | simp3 1138 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 < 𝐵) | |
3 | 1, 2 | syl6bi 252 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) → 𝐶 < 𝐵)) |
4 | 3 | 3impia 1117 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 class class class wbr 5103 (class class class)co 7353 ℝcr 11046 ℝ*cxr 11184 < clt 11185 (,)cioo 13256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-pre-lttri 11121 ax-pre-lttrn 11122 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-ioo 13260 |
This theorem is referenced by: iooshift 43692 icoopn 43695 iooiinicc 43712 iooltubd 43714 iooiinioc 43726 lptre2pt 43813 limcresiooub 43815 limcresioolb 43816 sinaover2ne0 44041 dvbdfbdioolem1 44101 dvbdfbdioolem2 44102 ioodvbdlimc1lem1 44104 ioodvbdlimc2lem 44107 fourierdlem27 44307 fourierdlem28 44308 fourierdlem40 44320 fourierdlem41 44321 fourierdlem46 44325 fourierdlem48 44327 fourierdlem49 44328 fourierdlem57 44336 fourierdlem59 44338 fourierdlem62 44341 fourierdlem64 44343 fourierdlem68 44347 fourierdlem73 44352 fourierdlem76 44355 fourierdlem78 44357 fourierdlem84 44363 fourierdlem90 44369 fourierdlem92 44371 fourierdlem97 44376 fourierdlem103 44382 fourierdlem104 44383 fourierdlem111 44390 sqwvfoura 44401 sqwvfourb 44402 fouriersw 44404 etransclem23 44430 qndenserrnbllem 44467 ioorrnopnlem 44477 ioorrnopnxrlem 44479 hspdifhsp 44789 hoiqssbllem1 44795 hoiqssbllem2 44796 hspmbllem2 44800 iunhoiioolem 44848 pimiooltgt 44883 pimdecfgtioo 44890 pimincfltioo 44891 smfaddlem1 44936 smfmullem1 44964 smfmullem2 44965 |
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