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| Mirrors > Home > MPE Home > Th. List > elicc1 | Structured version Visualization version GIF version | ||
| Description: Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.) |
| Ref | Expression |
|---|---|
| elicc1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-icc 13394 | . 2 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
| 2 | 1 | elixx1 13396 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝ*cxr 11294 ≤ 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-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 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-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xr 11299 df-icc 13394 |
| This theorem is referenced by: iccid 13432 iccleub 13442 iccgelb 13443 elicc2 13452 elicc4 13454 elxrge0 13497 lbicc2 13504 ubicc2 13505 difreicc 13524 cnblcld 24795 ovolf 25517 volivth 25642 itg2ge0 25770 itg2const2 25776 taylfvallem1 26398 tayl0 26403 radcnvcl 26460 radcnvle 26463 psercnlem1 26469 eliccelico 32779 xrdifh 32782 unitssxrge0 33899 esumle 34059 esumlef 34063 esumpinfsum 34078 voliune 34230 volfiniune 34231 ddemeas 34237 prob01 34415 elicc3 36318 ftc1cnnclem 37698 ftc1anc 37708 ftc2nc 37709 dvle2 42073 iocinico 43224 icoiccdif 45537 iblsplit 45981 iblspltprt 45988 itgspltprt 45994 fourierdlem1 46123 iccpartrn 47417 rrxsphere 48669 |
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