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| Mirrors > Home > MPE Home > Th. List > icossxr | Structured version Visualization version GIF version | ||
| Description: A closed-below, open-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.) | 
| Ref | Expression | 
|---|---|
| icossxr | ⊢ (𝐴[,)𝐵) ⊆ ℝ* | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ico 13393 | . 2 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
| 2 | 1 | ixxssxr 13399 | 1 ⊢ (𝐴[,)𝐵) ⊆ ℝ* | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊆ wss 3951 (class class class)co 7431 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 [,)cico 13389 | 
| 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-ne 2941 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-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 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-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-xr 11299 df-ico 13393 | 
| This theorem is referenced by: leordtvallem2 23219 leordtval2 23220 nmoffn 24732 nmofval 24735 nmogelb 24737 nmolb 24738 nmof 24740 icopnfhmeo 24974 elovolm 25510 ovolmge0 25512 ovolgelb 25515 ovollb2lem 25523 ovoliunlem1 25537 ovoliunlem2 25538 ovolscalem1 25548 ovolicc1 25551 ioombl1lem2 25594 ioombl1lem4 25596 uniioovol 25614 uniiccvol 25615 uniioombllem1 25616 uniioombllem2 25618 uniioombllem3 25620 uniioombllem6 25623 ply1degltdimlem 33673 esumpfinvallem 34075 esummulc1 34082 esummulc2 34083 mblfinlem3 37666 mblfinlem4 37667 ismblfin 37668 itg2gt0cn 37682 xralrple2 45365 icoub 45539 liminflelimsuplem 45790 elhoi 46557 hoidmvlelem5 46614 ovnhoilem1 46616 ovnhoilem2 46617 ovnhoi 46618 | 
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