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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 12420 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
2 | simp3 1132 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) → 𝐶 < 𝐵) | |
3 | 1, 2 | syl6bi 243 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) → 𝐶 < 𝐵)) |
4 | 3 | 3impia 1109 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 ∈ wcel 2145 class class class wbr 4786 (class class class)co 6792 ℝcr 10136 ℝ*cxr 10274 < clt 10275 (,)cioo 12379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-pre-lttri 10211 ax-pre-lttrn 10212 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-1st 7314 df-2nd 7315 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-ioo 12383 |
This theorem is referenced by: iooshift 40263 icoopn 40266 iooiinicc 40283 iooltubd 40285 iooiinioc 40297 lptre2pt 40386 limcresiooub 40388 limcresioolb 40389 sinaover2ne0 40593 dvbdfbdioolem1 40657 dvbdfbdioolem2 40658 ioodvbdlimc1lem1 40660 ioodvbdlimc2lem 40663 fourierdlem27 40864 fourierdlem28 40865 fourierdlem40 40877 fourierdlem41 40878 fourierdlem46 40882 fourierdlem48 40884 fourierdlem49 40885 fourierdlem57 40893 fourierdlem59 40895 fourierdlem60 40896 fourierdlem61 40897 fourierdlem62 40898 fourierdlem64 40900 fourierdlem68 40904 fourierdlem73 40909 fourierdlem76 40912 fourierdlem78 40914 fourierdlem84 40920 fourierdlem90 40926 fourierdlem92 40928 fourierdlem97 40933 fourierdlem103 40939 fourierdlem104 40940 fourierdlem111 40947 sqwvfoura 40958 sqwvfourb 40959 fouriersw 40961 etransclem23 40987 qndenserrnbllem 41027 ioorrnopnlem 41037 ioorrnopnxrlem 41039 hspdifhsp 41346 hoiqssbllem1 41352 hoiqssbllem2 41353 hspmbllem2 41357 iunhoiioolem 41405 pimiooltgt 41437 pimdecfgtioo 41443 pimincfltioo 41444 smfaddlem1 41487 smfmullem1 41514 smfmullem2 41515 |
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