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Mirrors > Home > MPE Home > Th. List > 0elunit | Structured version Visualization version GIF version |
Description: Zero is an element of the closed unit interval. (Contributed by Scott Fenton, 11-Jun-2013.) |
Ref | Expression |
---|---|
0elunit | ⊢ 0 ∈ (0[,]1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 11211 | . 2 ⊢ 0 ∈ ℝ | |
2 | 0le0 12308 | . 2 ⊢ 0 ≤ 0 | |
3 | 0le1 11732 | . 2 ⊢ 0 ≤ 1 | |
4 | elicc01 13438 | . 2 ⊢ (0 ∈ (0[,]1) ↔ (0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 ≤ 1)) | |
5 | 1, 2, 3, 4 | mpbir3an 1342 | 1 ⊢ 0 ∈ (0[,]1) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 class class class wbr 5146 (class class class)co 7403 ℝcr 11104 0cc0 11105 1c1 11106 ≤ cle 11244 [,]cicc 13322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-icc 13326 |
This theorem is referenced by: xrhmeo 24443 htpycom 24473 htpyid 24474 htpyco1 24475 htpyco2 24476 htpycc 24477 phtpy01 24482 phtpycom 24485 phtpyid 24486 phtpyco2 24487 phtpycc 24488 reparphti 24494 pcocn 24514 pcohtpylem 24516 pcoptcl 24518 pcopt 24519 pcopt2 24520 pcoass 24521 pcorevcl 24522 pcorevlem 24523 pi1xfrf 24550 pi1xfr 24552 pi1xfrcnvlem 24553 pi1xfrcnv 24554 pi1cof 24556 pi1coghm 24558 dvlipcn 25492 lgamgulmlem2 26513 ttgcontlem1 28121 brbtwn2 28142 axsegconlem1 28154 axpaschlem 28177 axcontlem7 28207 axcontlem8 28208 xrge0iifcnv 32850 xrge0iifiso 32852 xrge0iifhom 32854 cnpconn 34158 pconnconn 34159 txpconn 34160 ptpconn 34161 indispconn 34162 connpconn 34163 sconnpi1 34167 txsconnlem 34168 txsconn 34169 cvxpconn 34170 cvxsconn 34171 cvmliftlem14 34225 cvmlift2lem2 34232 cvmlift2lem3 34233 cvmlift2lem8 34238 cvmlift2lem12 34242 cvmlift2lem13 34243 cvmliftphtlem 34245 cvmliftpht 34246 cvmlift3lem1 34247 cvmlift3lem2 34248 cvmlift3lem4 34250 cvmlift3lem5 34251 cvmlift3lem6 34252 cvmlift3lem9 34255 gg-reparphti 35109 lcmineqlem12 40842 |
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