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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 13452 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 ≤ cle 11296 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-icc 13394 |
| This theorem is referenced by: elicc01 13506 sinbnd2 16218 cosbnd2 16219 iihalf1 24958 iihalf2 24961 elii1 24964 elii2 24965 xrhmeo 24977 oprpiece1res2 24983 pco0 25047 pcoval2 25049 pcoass 25057 vitalilem2 25644 vitali 25648 coseq00topi 26544 coseq0negpitopi 26545 sinq12ge0 26550 cosq14ge0 26553 cosordlem 26572 cosord 26573 cos11 26575 sinord 26576 recosf1o 26577 resinf1o 26578 efif1olem3 26586 argregt0 26652 argrege0 26653 argimgt0 26654 logimul 26656 cxpsqrtlem 26744 acosbnd 26943 log2ub 26992 emcllem7 27045 emgt0 27050 harmonicbnd3 27051 harmoniclbnd 27052 harmonicubnd 27053 harmonicbnd4 27054 logdivbnd 27600 pntpbnd2 27631 sin2h 37617 cos2h 37618 asin1half 42387 lhe4.4ex1a 44348 fourierdlem40 46162 fourierdlem62 46183 fourierdlem78 46199 fourierdlem111 46232 sqwvfoura 46243 sqwvfourb 46244 |
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