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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 13425 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
2 | simp3 1137 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 < 𝐵) | |
3 | 1, 2 | biimtrdi 253 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) → 𝐶 < 𝐵)) |
4 | 3 | 3impia 1116 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 ℝ*cxr 11292 < clt 11293 (,)cioo 13384 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ioo 13388 |
This theorem is referenced by: iooshift 45475 icoopn 45478 iooiinicc 45495 iooltubd 45497 iooiinioc 45509 lptre2pt 45596 limcresiooub 45598 limcresioolb 45599 sinaover2ne0 45824 dvbdfbdioolem1 45884 dvbdfbdioolem2 45885 ioodvbdlimc1lem1 45887 ioodvbdlimc2lem 45890 fourierdlem27 46090 fourierdlem28 46091 fourierdlem40 46103 fourierdlem41 46104 fourierdlem46 46108 fourierdlem48 46110 fourierdlem49 46111 fourierdlem57 46119 fourierdlem59 46121 fourierdlem62 46124 fourierdlem64 46126 fourierdlem68 46130 fourierdlem73 46135 fourierdlem76 46138 fourierdlem78 46140 fourierdlem84 46146 fourierdlem90 46152 fourierdlem92 46154 fourierdlem97 46159 fourierdlem103 46165 fourierdlem104 46166 fourierdlem111 46173 sqwvfoura 46184 sqwvfourb 46185 fouriersw 46187 etransclem23 46213 qndenserrnbllem 46250 ioorrnopnlem 46260 ioorrnopnxrlem 46262 hspdifhsp 46572 hoiqssbllem1 46578 hoiqssbllem2 46579 hspmbllem2 46583 iunhoiioolem 46631 pimiooltgt 46666 pimdecfgtioo 46673 pimincfltioo 46674 smfaddlem1 46719 smfmullem1 46747 smfmullem2 46748 |
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