![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12385 | . 2 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | 1 | elixx1 12388 | 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 196 ∧ wa 382 ∧ w3a 1071 ∈ wcel 2145 class class class wbr 4786 (class class class)co 6792 ℝ*cxr 10274 < clt 10275 ≤ cle 10276 [,)cico 12381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-iota 5994 df-fun 6033 df-fv 6039 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-xr 10279 df-ico 12385 |
This theorem is referenced by: elicod 12428 icogelb 12429 lbico1 12432 elico2 12441 icodisj 12503 ico01fl0 12827 addmodid 12925 leordtvallem2 21235 pnfnei 21244 mnfnei 21245 metustexhalf 22580 blval2 22586 metuel2 22589 iscfil2 23282 eliccelico 29876 elicoelioo 29877 xrdifh 29879 fsumrp0cl 30032 xrge0iifcnv 30316 esumpcvgval 30477 dnizeq0 32799 relowlssretop 33544 tan2h 33730 iocinico 38319 rfcnpre3 39710 icoltub 40249 icoiccdif 40265 iccelpart 41893 icceuelpart 41896 bgoldbtbndlem1 42217 bgoldbtbndlem2 42218 bgoldbtbndlem3 42219 bgoldbtbnd 42221 |
Copyright terms: Public domain | W3C validator |