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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 13374 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) | |
7 | 4, 5, 6 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
8 | 1, 2, 3, 7 | mpbir3and 1341 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 ∈ wcel 2105 class class class wbr 5148 (class class class)co 7412 ℝ*cxr 11254 < clt 11255 ≤ cle 11256 [,)cico 13333 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-xr 11259 df-ico 13337 |
This theorem is referenced by: fprodge1 15946 metustexhalf 24384 ply1degltel 33105 ply1degleel 33106 ply1degltlss 33107 ply1degltdimlem 33160 ply1degltdim 33161 absfico 44375 icoiccdif 44695 icoopn 44696 eliccnelico 44700 eliccelicod 44701 ge0xrre 44702 uzinico 44731 fsumge0cl 44747 limsupresico 44874 limsuppnfdlem 44875 limsupmnflem 44894 liminfresico 44945 limsup10exlem 44946 liminflelimsupuz 44959 xlimmnfvlem2 45007 icocncflimc 45063 fourierdlem41 45322 fourierdlem46 45326 fourierdlem48 45328 fouriersw 45405 fge0iccico 45544 sge0tsms 45554 sge0repnf 45560 sge0pr 45568 sge0iunmptlemre 45589 sge0rpcpnf 45595 sge0rernmpt 45596 sge0ad2en 45605 sge0xaddlem2 45608 voliunsge0lem 45646 meassre 45651 meaiuninclem 45654 omessre 45684 omeiunltfirp 45693 hoiprodcl 45721 hoicvr 45722 ovnsubaddlem1 45744 hoiprodcl3 45754 hoidmvcl 45756 hoidmv1lelem3 45767 hoidmvlelem3 45771 hoidmvlelem5 45773 hspdifhsp 45790 hoiqssbllem1 45796 hoiqssbllem2 45797 hspmbllem2 45801 volicorege0 45811 ovolval5lem1 45826 iunhoiioolem 45849 preimaicomnf 45885 mod42tp1mod8 46728 eenglngeehlnmlem2 47585 itscnhlinecirc02p 47632 |
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