![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 13365 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
2 | simp3 1139 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 < 𝐵) | |
3 | 1, 2 | syl6bi 253 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) → 𝐶 < 𝐵)) |
4 | 3 | 3impia 1118 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5149 (class class class)co 7409 ℝcr 11109 ℝ*cxr 11247 < clt 11248 (,)cioo 13324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-ioo 13328 |
This theorem is referenced by: iooshift 44235 icoopn 44238 iooiinicc 44255 iooltubd 44257 iooiinioc 44269 lptre2pt 44356 limcresiooub 44358 limcresioolb 44359 sinaover2ne0 44584 dvbdfbdioolem1 44644 dvbdfbdioolem2 44645 ioodvbdlimc1lem1 44647 ioodvbdlimc2lem 44650 fourierdlem27 44850 fourierdlem28 44851 fourierdlem40 44863 fourierdlem41 44864 fourierdlem46 44868 fourierdlem48 44870 fourierdlem49 44871 fourierdlem57 44879 fourierdlem59 44881 fourierdlem62 44884 fourierdlem64 44886 fourierdlem68 44890 fourierdlem73 44895 fourierdlem76 44898 fourierdlem78 44900 fourierdlem84 44906 fourierdlem90 44912 fourierdlem92 44914 fourierdlem97 44919 fourierdlem103 44925 fourierdlem104 44926 fourierdlem111 44933 sqwvfoura 44944 sqwvfourb 44945 fouriersw 44947 etransclem23 44973 qndenserrnbllem 45010 ioorrnopnlem 45020 ioorrnopnxrlem 45022 hspdifhsp 45332 hoiqssbllem1 45338 hoiqssbllem2 45339 hspmbllem2 45343 iunhoiioolem 45391 pimiooltgt 45426 pimdecfgtioo 45433 pimincfltioo 45434 smfaddlem1 45479 smfmullem1 45507 smfmullem2 45508 |
Copyright terms: Public domain | W3C validator |