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Mirrors > Home > MPE Home > Th. List > elico1 | Structured version Visualization version GIF version |
Description: Membership in a closed-below, open-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
elico1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ico 12732 | . 2 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | 1 | elixx1 12735 | 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 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 [,)cico 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-xr 10668 df-ico 12732 |
This theorem is referenced by: elicod 12775 icogelb 12776 lbico1 12779 elico2 12789 icodisj 12854 ico01fl0 13184 addmodid 13282 leordtvallem2 21816 pnfnei 21825 mnfnei 21826 blval2 23169 metuel2 23172 iscfil2 23870 eliccelico 30526 elicoelioo 30527 xrdifh 30529 fsumrp0cl 30729 xrge0iifcnv 31286 esumpcvgval 31447 dnizeq0 33927 relowlssretop 34780 tan2h 35049 iocinico 40162 rfcnpre3 41662 icoltub 42145 icoiccdif 42161 iccelpart 43950 icceuelpart 43953 bgoldbtbndlem1 44323 bgoldbtbndlem2 44324 bgoldbtbndlem3 44325 bgoldbtbnd 44327 |
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