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Mirrors > Home > MPE Home > Th. List > elicod | Structured version Visualization version GIF version |
Description: Membership in a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
elicod.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
elicod.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
elicod.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
elicod.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
elicod.5 | ⊢ (𝜑 → 𝐶 < 𝐵) |
Ref | Expression |
---|---|
elicod | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elicod.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
2 | elicod.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶) | |
3 | elicod.5 | . 2 ⊢ (𝜑 → 𝐶 < 𝐵) | |
4 | elicod.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
5 | elicod.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
6 | elico1 12978 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) | |
7 | 4, 5, 6 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
8 | 1, 2, 3, 7 | mpbir3and 1344 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1089 ∈ wcel 2110 class class class wbr 5053 (class class class)co 7213 ℝ*cxr 10866 < clt 10867 ≤ cle 10868 [,)cico 12937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-xr 10871 df-ico 12941 |
This theorem is referenced by: fprodge1 15557 metustexhalf 23454 absfico 42431 icoiccdif 42737 icoopn 42738 eliccnelico 42742 eliccelicod 42743 ge0xrre 42744 uzinico 42773 fsumge0cl 42789 limsupresico 42916 limsuppnfdlem 42917 limsupmnflem 42936 liminfresico 42987 limsup10exlem 42988 liminflelimsupuz 43001 xlimmnfvlem2 43049 icocncflimc 43105 fourierdlem41 43364 fourierdlem46 43368 fourierdlem48 43370 fouriersw 43447 fge0iccico 43583 sge0tsms 43593 sge0repnf 43599 sge0pr 43607 sge0iunmptlemre 43628 sge0rpcpnf 43634 sge0rernmpt 43635 sge0ad2en 43644 sge0xaddlem2 43647 voliunsge0lem 43685 meassre 43690 meaiuninclem 43693 omessre 43723 omeiunltfirp 43732 hoiprodcl 43760 hoicvr 43761 ovnsubaddlem1 43783 hoiprodcl3 43793 hoidmvcl 43795 hoidmv1lelem3 43806 hoidmvlelem3 43810 hoidmvlelem5 43812 hspdifhsp 43829 hoiqssbllem1 43835 hoiqssbllem2 43836 hspmbllem2 43840 volicorege0 43850 ovolval5lem1 43865 iunhoiioolem 43888 preimaicomnf 43921 mod42tp1mod8 44727 eenglngeehlnmlem2 45757 itscnhlinecirc02p 45804 |
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