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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 12804 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 ≤ cle 10678 [,]cicc 12744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-icc 12748 |
This theorem is referenced by: elicc01 12857 sinbnd2 15537 cosbnd2 15538 iihalf1 23537 iihalf2 23539 elii1 23541 elii2 23542 xrhmeo 23552 oprpiece1res2 23558 pco0 23620 pcoval2 23622 pcoass 23630 vitalilem2 24212 vitali 24216 coseq00topi 25090 coseq0negpitopi 25091 sinq12ge0 25096 cosq14ge0 25099 cosordlem 25117 cosord 25118 cos11 25119 sinord 25120 recosf1o 25121 resinf1o 25122 efif1olem3 25130 argregt0 25195 argrege0 25196 argimgt0 25197 logimul 25199 cxpsqrtlem 25287 acosbnd 25480 log2ub 25529 emcllem7 25581 emgt0 25586 harmonicbnd3 25587 harmoniclbnd 25588 harmonicubnd 25589 harmonicbnd4 25590 logdivbnd 26134 pntpbnd2 26165 sin2h 34884 cos2h 34885 lhe4.4ex1a 40668 fourierdlem40 42439 fourierdlem62 42460 fourierdlem78 42476 fourierdlem111 42509 sqwvfoura 42520 sqwvfourb 42521 |
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