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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 12802 | . 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 5066 (class class class)co 7156 ℝcr 10536 ≤ cle 10676 [,]cicc 12742 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-icc 12746 |
This theorem is referenced by: elicc01 12855 sinbnd2 15535 cosbnd2 15536 iihalf1 23535 iihalf2 23537 elii1 23539 elii2 23540 xrhmeo 23550 oprpiece1res2 23556 pco0 23618 pcoval2 23620 pcoass 23628 vitalilem2 24210 vitali 24214 coseq00topi 25088 coseq0negpitopi 25089 sinq12ge0 25094 cosq14ge0 25097 cosordlem 25115 cosord 25116 cos11 25117 sinord 25118 recosf1o 25119 resinf1o 25120 efif1olem3 25128 argregt0 25193 argrege0 25194 argimgt0 25195 logimul 25197 cxpsqrtlem 25285 acosbnd 25478 log2ub 25527 emcllem7 25579 emgt0 25584 harmonicbnd3 25585 harmoniclbnd 25586 harmonicubnd 25587 harmonicbnd4 25588 logdivbnd 26132 pntpbnd2 26163 sin2h 34897 cos2h 34898 lhe4.4ex1a 40681 fourierdlem40 42452 fourierdlem62 42473 fourierdlem78 42489 fourierdlem111 42522 sqwvfoura 42533 sqwvfourb 42534 |
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