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Mirrors > Home > MPE Home > Th. List > 1elunit | Structured version Visualization version GIF version |
Description: One is an element of the closed unit interval. (Contributed by Scott Fenton, 11-Jun-2013.) |
Ref | Expression |
---|---|
1elunit | ⊢ 1 ∈ (0[,]1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10494 | . 2 ⊢ 1 ∈ ℝ | |
2 | 0le1 11017 | . 2 ⊢ 0 ≤ 1 | |
3 | 1le1 11122 | . 2 ⊢ 1 ≤ 1 | |
4 | elicc01 12708 | . 2 ⊢ (1 ∈ (0[,]1) ↔ (1 ∈ ℝ ∧ 0 ≤ 1 ∧ 1 ≤ 1)) | |
5 | 1, 2, 3, 4 | mpbir3an 1334 | 1 ⊢ 1 ∈ (0[,]1) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2083 class class class wbr 4968 (class class class)co 7023 ℝcr 10389 0cc0 10390 1c1 10391 ≤ cle 10529 [,]cicc 12595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-po 5369 df-so 5370 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-icc 12599 |
This theorem is referenced by: iccpnfcnv 23235 htpycom 23267 htpyid 23268 htpyco1 23269 htpyco2 23270 htpycc 23271 phtpy01 23276 phtpycom 23279 phtpyid 23280 phtpyco2 23281 phtpycc 23282 reparphti 23288 pco1 23306 pcohtpylem 23310 pcoptcl 23312 pcopt 23313 pcopt2 23314 pcoass 23315 pcorevcl 23316 pcorevlem 23317 pi1xfrf 23344 pi1xfr 23346 pi1xfrcnvlem 23347 pi1xfrcnv 23348 pi1cof 23350 pi1coghm 23352 dvlipcn 24278 leibpi 25206 lgamgulmlem2 25293 ttgcontlem1 26358 axpaschlem 26413 iistmd 30758 xrge0iif1 30794 xrge0iifmhm 30795 cnpconn 32087 pconnconn 32088 txpconn 32089 ptpconn 32090 indispconn 32091 connpconn 32092 txsconnlem 32097 txsconn 32098 cvxpconn 32099 cvxsconn 32100 cvmliftphtlem 32174 cvmlift3lem2 32177 cvmlift3lem4 32179 cvmlift3lem5 32180 cvmlift3lem6 32181 cvmlift3lem9 32184 k0004val0 40010 |
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