![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elicc01 | Structured version Visualization version GIF version |
Description: Membership in the closed real interval between 0 and 1, also called the closed unit interval. (Contributed by AV, 20-Aug-2022.) |
Ref | Expression |
---|---|
elicc01 | ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 11266 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1re 11264 | . 2 ⊢ 1 ∈ ℝ | |
3 | 1, 2 | elicc2i 13444 | 1 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1084 ∈ wcel 2099 class class class wbr 5153 (class class class)co 7424 ℝcr 11157 0cc0 11158 1c1 11159 ≤ cle 11299 [,]cicc 13381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-i2m1 11226 ax-1ne0 11227 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-icc 13385 |
This theorem is referenced by: elunitrn 13498 0elunit 13500 1elunit 13501 divelunit 13525 lincmb01cmp 13526 iccf1o 13527 rpnnen2lem12 16227 blcvx 24805 iirev 24941 iihalf2 24946 elii2 24950 iimulcl 24951 iccpnfhmeo 24961 xrhmeo 24962 lebnumii 24983 htpycc 24997 pcocn 25035 pcohtpylem 25037 pcopt 25040 pcopt2 25041 pcoass 25042 pcorevlem 25044 vitalilem2 25629 abelth2 26472 chordthmlem4 26863 leibpi 26970 jensenlem2 27016 lgamgulmlem2 27058 ttgcontlem1 28818 brbtwn2 28839 ax5seglem1 28862 ax5seglem2 28863 ax5seglem3 28865 ax5seglem5 28867 ax5seglem6 28868 ax5seglem9 28871 ax5seg 28872 axbtwnid 28873 axpaschlem 28874 axpasch 28875 axcontlem2 28899 axcontlem4 28901 axcontlem7 28904 stge0 32157 stle1 32158 strlem3a 32185 elunitge0 33714 unitdivcld 33716 xrge0iifiso 33750 xrge0iifhom 33752 resconn 35074 snmlff 35157 poimirlem29 37350 poimirlem30 37351 poimirlem31 37352 poimirlem32 37353 i0oii 48253 io1ii 48254 |
Copyright terms: Public domain | W3C validator |