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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 10835 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1re 10833 | . 2 ⊢ 1 ∈ ℝ | |
3 | 1, 2 | elicc2i 13001 | 1 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ w3a 1089 ∈ wcel 2110 class class class wbr 5053 (class class class)co 7213 ℝcr 10728 0cc0 10729 1c1 10730 ≤ cle 10868 [,]cicc 12938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-i2m1 10797 ax-1ne0 10798 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-icc 12942 |
This theorem is referenced by: elunitrn 13055 0elunit 13057 1elunit 13058 divelunit 13082 lincmb01cmp 13083 iccf1o 13084 rpnnen2lem12 15786 blcvx 23695 iirev 23826 iihalf2 23830 elii2 23833 iimulcl 23834 iccpnfhmeo 23842 xrhmeo 23843 lebnumii 23863 htpycc 23877 pcocn 23914 pcohtpylem 23916 pcopt 23919 pcopt2 23920 pcoass 23921 pcorevlem 23923 vitalilem2 24506 abelth2 25334 chordthmlem4 25718 leibpi 25825 jensenlem2 25870 lgamgulmlem2 25912 ttgcontlem1 26976 brbtwn2 26996 ax5seglem1 27019 ax5seglem2 27020 ax5seglem3 27022 ax5seglem5 27024 ax5seglem6 27025 ax5seglem9 27028 ax5seg 27029 axbtwnid 27030 axpaschlem 27031 axpasch 27032 axcontlem2 27056 axcontlem4 27058 axcontlem7 27061 stge0 30305 stle1 30306 strlem3a 30333 elunitge0 31563 unitdivcld 31565 xrge0iifiso 31599 xrge0iifhom 31601 resconn 32921 snmlff 33004 poimirlem29 35543 poimirlem30 35544 poimirlem31 35545 poimirlem32 35546 i0oii 45886 io1ii 45887 |
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