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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 12782 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) | |
7 | 4, 5, 6 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
8 | 1, 2, 3, 7 | mpbir3and 1338 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 [,)cico 12741 |
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-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-xr 10679 df-ico 12745 |
This theorem is referenced by: fprodge1 15349 metustexhalf 23166 absfico 41530 icoiccdif 41849 icoopn 41850 eliccnelico 41854 eliccelicod 41855 ge0xrre 41856 uzinico 41885 fsumge0cl 41903 limsupresico 42030 limsuppnfdlem 42031 limsupmnflem 42050 liminfresico 42101 limsup10exlem 42102 liminflelimsupuz 42115 xlimmnfvlem2 42163 icocncflimc 42221 fourierdlem41 42482 fourierdlem46 42486 fourierdlem48 42488 fouriersw 42565 fge0iccico 42701 sge0tsms 42711 sge0repnf 42717 sge0pr 42725 sge0iunmptlemre 42746 sge0rpcpnf 42752 sge0rernmpt 42753 sge0ad2en 42762 sge0xaddlem2 42765 voliunsge0lem 42803 meassre 42808 meaiuninclem 42811 omessre 42841 omeiunltfirp 42850 hoiprodcl 42878 hoicvr 42879 ovnsubaddlem1 42901 hoiprodcl3 42911 hoidmvcl 42913 hoidmv1lelem3 42924 hoidmvlelem3 42928 hoidmvlelem5 42930 hspdifhsp 42947 hoiqssbllem1 42953 hoiqssbllem2 42954 hspmbllem2 42958 volicorege0 42968 ovolval5lem1 42983 iunhoiioolem 43006 preimaicomnf 43039 mod42tp1mod8 43816 eenglngeehlnmlem2 44774 itscnhlinecirc02p 44821 |
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