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Mirrors > Home > MPE Home > Th. List > elicc2i | Structured version Visualization version GIF version |
Description: Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.) |
Ref | Expression |
---|---|
elicc2i.1 | ⊢ 𝐴 ∈ ℝ |
elicc2i.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
elicc2i | ⊢ (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elicc2i.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | elicc2i.2 | . 2 ⊢ 𝐵 ∈ ℝ | |
3 | elicc2 12790 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ w3a 1084 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 ≤ cle 10665 [,]cicc 12729 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-icc 12733 |
This theorem is referenced by: elicc01 12844 sinbnd2 15527 cosbnd2 15528 iihalf1 23536 iihalf2 23538 elii1 23540 elii2 23541 xrhmeo 23551 oprpiece1res2 23557 pco0 23619 pcoval2 23621 pcoass 23629 vitalilem2 24213 vitali 24217 coseq00topi 25095 coseq0negpitopi 25096 sinq12ge0 25101 cosq14ge0 25104 cosordlem 25122 cosord 25123 cos11 25125 sinord 25126 recosf1o 25127 resinf1o 25128 efif1olem3 25136 argregt0 25201 argrege0 25202 argimgt0 25203 logimul 25205 cxpsqrtlem 25293 acosbnd 25486 log2ub 25535 emcllem7 25587 emgt0 25592 harmonicbnd3 25593 harmoniclbnd 25594 harmonicubnd 25595 harmonicbnd4 25596 logdivbnd 26140 pntpbnd2 26171 sin2h 35047 cos2h 35048 lhe4.4ex1a 41033 fourierdlem40 42789 fourierdlem62 42810 fourierdlem78 42826 fourierdlem111 42859 sqwvfoura 42870 sqwvfourb 42871 |
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